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PUBLISHED  BY 

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MATHEMATICAL    MONOGRAPHS 

EDITED   BY 

MANSFIELD  MERRIMAN  AND  ROBERT  S.  WOODWARD 


No.  21 

THE  DYNAMICS  OF  THE 
AIRPLANE 


BY 


KENNETH  P.  WILLIAMS,  PH.D. 

ASSOCIATE  PROFESSOR  OF  MATHEMATICS 
INDIANA  UNIVERSITY 


NEW  YORK 

JOHN  WILEY  &  SONS,  INC. 

LONDON:    CHAPMAN  &  HALL,  LIMITED 

1921 


W  5" 


Library 


COPYRIGHT,  1921 

BY 
K.  P.  WILLIAMS 


PRESS  Of 

RRAUNWORTH    li    CO. 

BOOK  MANUFACTURERS 

UROOKUYN.    N.   Y. 


PREFACE 


IT  was  the  good  fortune  of  the  author  to  attend  the  University 
of  Paris  during  the  spring  semester  of  1919.  One  of  the  special 
courses  which  the  French  authorities,  with  their  characteristic 
hospitality,  arranged  for  the  large  number  of  students  from  the 
American  army,  was  a  course  in  aerodynamics,  given  by  Professor 
Marchis.  The  comprehensive  knowledge  that  Professor  Marchis 
possessed  of  all  branches  of  the  new  science  of  aeronautics,  the 
inestimable  value  of  his  advice  to  the  French  Republic  during 
the  war,  the  interest  he  took  in  his  rather  unusual  class,  could  not 
fail  to  be  an  inspiration. 

This  book  is  an  outgrowth  of  those  parts  of  Professor  Marchis' 
lectures  that  were  of  particular  interest  to  the  author.  It  is  in 
no  sense  a  complete  treatise  on  aviation.  Questions  of  design 
and  construction  are  passed  over  with  bare  mention.  The  book 
is  intended  for  students  of  mathematics  and  physics  who  are 
attracted  by  the  dynamical  aspect  of  aviation.  The  problems 
presented  by  the  motion  of  an  airplane  are  novel  and  fasci- 
nating. They  vary  from  the  most  pleasing  simplicity  to  the 
most  stimulating  difficulty.  The  question  of  stability,  partic- 
ularly, exhibits  at  the  same  time  the  elegance  and  the  power 
of  analysis,  and  shows  the  adaptability  of  some  of  the  general 
developments  in  dynamics.  The  field  is  assuredly  a  fruitful 
one  of  study,  and  increasing  demands  will  be  put  upon  the 
mathematician  as  the  science  of  aviation  continues  its  rapid 
development.  The  mathematician  can  well  own  a  sense  of  pride 
that  he  had  already  at  hand,  in  the  developments  inaugurated 
by  Euler  and  Routh,  a  means  of  dealing  accurately  with  the 
question  of  stability,  that  plays  so  fundamental  a  role  in  the 
science  of  flying. 

iii 


447195 


IV  PREFACE 

The  treatment  in  the  text  is  for  the  most  part  elementary. 
The  last  chapter  alone  demands  of  the  student  familiarity  with 
more  advanced  dynamical  methods.  In  the  treatment  of  descent 
a  slight  digression  is  made  to  consider  in  part  the  nature  of  the 
solution  of  a  system  of  two  differential  equations.  This  was  done 
in  order  not  to  completely  evade  what  seems  a  problem  of  con- 
siderable difficulty.  It  might  seem  that  a  treatment  of  the 
propeller  should  not  find  a  place  in  a  book  with  the  purpose  of 
this  one.  No  student  of  mathematics,  however,  could  fail  to 
own  a  curiosity  as  to  a  propeller's  action,  and  it  is  hoped  the  dis- 
cussion, while  not  complete,  will  at  least  serve  as  a  sufficient 
introduction. 

The  various  curves  in  the  text  were  plotted  by  Mr.  R.  W. 
Smith,  a  former  student  in  this  university.  The  author  is 
further  indebted  to  the  Smithsonian  Institution  for  permission 
to  use  Figs.  12  and  49. 

In  addition  to  the  various  books  that  are  referred  to  in  the 
text  the  author  has  made  use  of  his  notes  of  the  lectures  of  Pro- 
fessor Marchis,  translated  into  English  by  Madame  Ciolkowska, 
who  rendered  most  valuable  aid  as  an  interpreter  for  those  who 
understood  and  spoke  the  language  of  Professor  Marchis  only 
with  difficulty. 

•K.  P.  WILLIAMS. 

Indiana  University, 
July,  1920. 


CONTENTS 


CHAPTER  I.    THE  PLANE  AND  CAMBERED  SURFACE 

ARTICLE  PAGE 

i,  2.   PRELIMINARY  CONSIDERATIONS.  . ; 1-3 

3.  PRESSURE  ON  A  PLANE 4 

4.  THE  INCLINED  PLANE 5 

5.  THE  CENTER  OF  PRESSURE 6 

6.  ASPECT  RATIO 6 

7.  THE  CAMBERED  WING 7 

8.  CHARACTERISTIC  CURVES  FOR  A  GIVEN  WING 8,  9 

9.  POLAR  DIAGRAM 10 

10.  EFFECT  OF  VARIATION  OF  WING.  ELEMENTS n 

11.  PRESSURE  OVER  THE  WING n 

12.  CENTER  OF  PRESSURE 12 

13.  RELATION  BETWEEN  Ky  AND  Kz 13 

14.  THE  BIPLANE 14 

15.  BODY  RESISTANCE 15 

16.  EXPERIMENTS  ON  COMPLETE  MODEL 15 

CHAPTER  II.     STRAIGHT  HORIZONTAL  FLIGHT 

17.  PRELIMINARY  CONSIDERATIONS 18 

18.  HORIZONTAL  FLIGHT 19 

19.  THE  VELOCITY 20 

20.  LANDING  SPEED 20 

21.  EFFECT  OF  ALTITUDE 21 

22.  THE  EFFORT  OF  TRACTION 22 

23.  OPT  MUM  ANGLE 22 

24.  FINENESS 23 

25.  OPTIMUM  ANGLE,  CONTINUED 24 

26,  27.  USEFUL  POWER 25 

28,  29.   ECONOMICAL  ANGLE 26-28 

30.  GENERAL  CONSIDERATIONS 29 

CHAPTER  III.     DESCENT  AND  ASCENT 
i.  DESCENT 

31.  PRELIMINARY  CONSIDERATIONS  CONCERNING  DESCENT 31 

32.  EQUATIONS  OF  MOTION 31 

v 


VI  CONTENTS 

ARTICLE  PAGE 

33, 34.  RECTILINEAR  DESCENT 33~35 

35.   GENERAL  RESULTS  CONCERNING  DESCENT 36 

36-38.  DESCENT  CONSIDERING  AIR  DENSITY  CONSTANT 38-42 

2.  ASCENT 

39.  PRELIMINARY  CONSIDERATIONS  CONCERNING  ASCENT 43 

40.  EQUATIONS  OF  MOTION  FOR  ASCENT 43 

41.  VELOCITY  ALONG  PATH 44 

42.  THE  FORCE  OF  TRACTION 44 

43.  THE  POWER  NECESSARY 46 

44.  THE  VERTICAL  VELOCITY 47 

45.  GENERAL  CONSIDERATIONS 48 

46.  EXPERIMENTAL  LAW  OF  VERTICAL  VELOCITY 49 

47.  TIME  OF  ASCENT 50 

48.  DETERMINATION  OF  THE  CEILING 50 


CHAPTER  IV.     CIRCULAR  FLIGHT 

1.  HORIZONTAL  TURNS, 

49.  GENERAL  CONSIDERATIONS 52 

50.  EQUATIONS  OF  MOTION 53 

51.  VELOCITY  AND  INCLINATION 54 

52.  THE  TRACTION  AND  POWER 55 

53.  ACTION  OF  CONTROLS 56 

2.  CIRCULAR  DESCENT 

54.  GENERAL  CONSIDERATIONS 58 

55.  EQUATIONS  OF  MOTION 59 

56.  RELATIONS  BETWEEN  ANGLES  USED 60 

57.  OTHER  FORM  OF  EQUATIONS  OF  MOTION 61 

58.  IDENTITY  OF  Two  FORMS  OF  EQUATIONS  OF  MOTION 62 

59.  DETERMINATION  OF  ANGLES  SPECIFYING  MOTION 63-65 

60.  THE  VELOCITY 66 

CHAPTER  V.    THE  PROPELLER 

61-64.  GENERAL  CONSIDERATIONS 69,  70 

65.  GEOMETRICAL  PITCH 71 

66-68.  THE  THRUST  AND  POWER ...   74-76 

69.  EFFICIENCY 77 

70.  EFFECT  OF  ALTITUDE 77 

71,  72.  GRAPHS  OF  PROPELLER  COEFFICIENTS 78 

73.  MOTOR  DIAGRAM 8 1 

74-78.  ADAPTATION  OF  MOTOR- PROPELLER  GROUP  TO  MACHINE 82-85 


CONTENTS  Vll 


CHAPTER  VI.     PERFORMANCE 

i.  CEILING 
ARTICLE  PAGE 

79.  GENERAL  CONSIDERATIONS 86 

80.  DETERMINATION  OF  CEILING 86 

81.  SUPERCHARGE 89 

2.  RADIUS  OF  ACTION 

82.  DETERMINATION  OF  DISTANCE  A  MACHINE  CAN  FLY 90 

83.  PROBLEM  OF  RETURN  JOURNEY 93 

CHAPTER  VII.     STABILITY  AND  CONTROLLABILITY 

84-88.  PRELIMINARY  CONSIDERATIONS 94-96 

89.  STATIC  AND  DYNAMIC  STABILITY 96 

90.  PITCHING,  ROLLING,  AND  YAWING 97 

LONGITUDINAL  STABILITY 

91.  PRELIMINARY  CONSIDERATIONS 97 

92.  METACENTRIC  CURVE 98 

93.  THE  TAIL  PLANE 99 

94.  THE  ELEVATOR 100 

95.  GENERAL  CONSIDERATIONS 102 

STABILITY  IN  ROLLING 

96.  PRELIMINARY  CONSIDERATIONS 102 

97.  DIHEDRAL 103 

98.  CONTROLLABILITY,  AILERONS 104 

LATERAL  STABILITY 

99.  FIN,  RUDDER 194 

100.  ACTION  OF  RUDDER 105 

101.  CONNECTION  BETWEEN  YAWING  AND  ROLLING 105 

102.  SPIRAL  INSTABILITY .   106 


CHAPTER  VIII.     STABILITY  (CONTINUED) 

103.  METHOD  OF  BRYAN 107 

104, 105.  MOVING  AXES 108 

106.  ANGULAR  VELOCITIES  AND  MOMENTA 109 

107, 108.  ORIENTATION 110-112 

109.  EQUATIONS  OF  MOTIONS , 113 

no.  STEADY  MOTION 114 


Vlll  CONTENTS 

ARTICLE  PACK 

in,  112.  SYMMETRIC  AND  ASYMMETRIC  OSCILLATIONS 115-117 

113.  THE  FUNDAMENTAL  QUARTIC  EQUATIONS 118 

114.  THE  CONDITIONS  OF  STABILITY 120 

115.  THE  FORM  OF  THE  COEFFICIENTS 120 

116.  THE  DETERMINATION  OF  THE  COEFFICIENTS 121 

117.  EXAMPLE 122 

118.  DEPENDENCE  OF  STABILITY  ON  SPEED 123 

119.  FACTORED  FORM  OF  QUARTIC  EQUATIONS 123 

120.  EXAMPLE 1 24 

121.  EFFECT  OF  GUSTS.     GENERAL  CONSIDERATIONS 125 

122.  EQUATIONS  FOR  TREATING  GUSTS 126 

123, 124.  EXAMPLE 128, 129 

APPENDIX 

1-3.  TRANSFORMATION  OF  UNITS 131-133 

4.  TABLE  OF  AIR  PRESSURE  AND  DENSITY 134 

5.  TABLE  FOR  VELOCITY  TRANSFORMATION 135 

6.  REFERENCES 135 


THE  DYNAMICS  OF  THE  AIRPLANE 


CHAPTER  I 
THE  PLANE  AND   CAMBERED  SURFACE 

1.  THE  possibility  of  aerial  navigation  depends  upon  the 
solution  of  two  problems,  the  problem  of  sustentation  and 
the  problem  of  propulsion.  At  the  very  outset  two  distinct 
courses  are  therefore  open.  We  can  look  upon  the  problems 
as  entirely  separated  from  each  other,  or  we  can  regard  them 
as  essentially  connected. 

In  the  first  case  we  look  for  separate  solutions,  solving  first 
the  problem  of  sustentation,  and  then,  with  this  successfully 
disposed  of,  search  for  a  means  of  propulsion.  This  course 
is  historically  the  older  and  it  is  the  simpler,  for  it  meets 
the  difficulties  one  at  a  time.*  A  balloon  filled  with  a  light 

*Early  literature  abounds  with  mythical  accounts  of  the  flights  of  legendary 
heroes  equipped  birdlike  with  wings.  Among  those  who  seriously  studied  the 
question  of  flight,  and  actually  designed  machines  with  wings  to  be  attached  to  a 
person  and  driven  by  his  own  muscular  power,  Leonardo  da  Vinci,  the  renowned 
artist,  occupies  the  first  place.  The  first  instance  of  people  actually  ascending 
from  the  earth  took  place  November  21,  1783,  at  Paris.  The  apparatus  was  a 
balloon  constructed  by  Stephen  Montgolfier,  and  the  flight  covered  five  miles. 
The  construction  of  crude  dirigibles  followed  within  a  few  months.  The  first 
instance  of  partial  sustentation  without  the  use  of  gas  occurred  at  this  same  period. 
On  December  26,  1784,  Sebastian  Lenormand  descended  from  the  tower  of  the 
Montpelier  Observatory  by  means  of  two  small  parachutes,  the  idea  of  the  para- 
chute being  due  to  da  Vinci.  Successful  efforts  with  gliders  were  made  in  the  last 
years  of  the  igth  century.  Modern  aviation  dates  from  1903,  when  the  Wright 
brothers  first  constructed  a  machine,  equipped  with  engines,  which  could  actually 
rise  from  a  level  field,  without  the  assistance  of  air  currents,  and  make  flights  con- 
trolled by  the  pilot.  For  the  history  of  aeronautics  see  Albert  F.  Zahm,  "Aerial 
Navigation,"  D.  Appleton,  &  Company,  New  York  and  London,  1911. 


2  '  ftiE  DYNAMICS  -O>  THE   AIRPLANE 

gas,  such  as  hydtogeri ^ or  4tefttinV  affords  -a  means  of  susten- 
tation.  But  aerial  navigation  means  far  more  than  the  ability 
to  stay  aloft,  and  a  craft  which  can  travel  only  as  the  wind 
blows  it,  can  serve  few  purposes  other  than  that  of  furnishing 
amusement.  The  question  of  equipping  a  balloon  with  engines 
and  a  means  of  propulsion,  of  traveling  in  a  desired  direction 
with  a  velocity  within  our  control,  of  maintaining  a  desired 
altitude,  of  rising  and  landing,  must  be  answered  before  we 
can  say  the  balloon  has  furnished  a  means  of  aerial  navigation. 
It  is  only  since  the  development  of  the  gasoline  motor  that  this 
has  been  possible  on  any  extensive  scale. 

The  second  method  of  attacking  the  problem  seeks  to  solve 
simultaneously  the  problems  of  sustentation  and  propulsion. 
The  possibility  of  propulsion  must  now  come  first,  for  susten- 
tation will  be  obtained  from  the  motion.  It  is  then  evident 
that  this  method,  even  more  than  that  of  the  balloon,  had  to 
await  the  perfection  of  a  source  of  energy  such  as  the  gasoline 
engine.  It  is  only  with  aircraft  of  this  second  sort  that  we  are 
concerned.  Such  a  machine  is  called  an  airplane,  or  aeroplane. 
We  shall  adopt  the  first  term.  Both  names  are  suggested  by 
the  fundamental  role  played  by  surfaces  approximately  plane 
with  which  the  machine  is  provided.  The  air  reaction  on  these 
surfaces,  produced  by  the  motion  of  the  machine,  furnishes  the 
sustentation.  The  complete  machine  will  consist  of  other  mem- 
bers, and  we  can  divide  it  into  five  distinct  parts:  the  sustain- 
ing surfaces,  the  stabilizing  and  controlling  surfaces,  the  motor- 
propeller  group,  the  body,  or  fuselage,  with  its  place  for  pilot, 
passengers  and  freight,  and  the  landing  gear.  To  these  main 
parts  must,  of  course,  be  added  the  various  elements  of  con- 
struction by  which  the  different  parts  are  united  and  the 
requisite  strength  given  to  the  complete  machine. 

We  shall  not  give  a  discussion  of  the  complete  construction 
of  an  airplane,  but  limit  ourselves  to  those  features  which 
are  necessary  for  a  comprehension  of  the  dynamical  problems 
which  we  shall  study. 

2.  The  principles  that  govern  the  construction  of  an  air- 
plane, the  phenomena  that  operate  during  its  flight  and  deter- 


THE  PLANE  AND  CAMBERED   SURFACE  3 

mine  its  behavior,  are  derived  from  an  understanding  of  the 
laws  concerning  the  effect  of  the  wind  upon  flat  and  curved 
surfaces.  We  can  try  to  determine  these  laws  in  two  ways, 
mathematically  or  experimentally. 

In  the  mathematical  method  we  begin  with  the  principles 
and  equations  of  hydromechanics.  We  then  see  if  we  can  cal- 
culate the  pressure  that  a  current  of  air,  moving  with  a  certain 
velocity,  will  exert  upon,  for  instance,  a  rectangular  plane 
surface.  We  must  be  able  to  do  this  for  different  inclinations 
of  the  plane  to  the  air  stream.  The  problem  is  one  of  great 
complexity.  In  order  to  construct  differential  equations  that 
will  exhibit  the  phenomena,  and  in  order  to  integrate  these, 
we  must  make  assumptions  that  lead  our  results  to  differ  from 
carefully  measured  observations.  For  instance,  we  may 
assume  that  the  air  is  a  perfect  fluid,  that  is,  that  it  is  neither 
viscous  nor  compressible.  The  last  assumption  seems  to  be 
justified  for  the  range  of  velocities  occurring  in  aeronautical 
work,  but  the  assumption  as  to  the  viscosity  vitiates  our 
results  when  we  apply  them  to  actual  problems.  Even  with 
the  assumption  of  an  ideal  fluid  it  is  difficult  to  handle  the 
equations  involved.  While  we  can  in  certain  instances  obtain 
information  as  to  how  the  air  streams  around  obstructing 
objects,  and  how  it  behaves  in  their  vicinity,  the  results  are 
not  such  as  to  make  this  a  simple  or  satisfactory  method  of 
attacking  the  problem.* 

The  laws  concerning  air  reaction  are  determined  experi- 
mentally in  a  wind  tunnel,  f  A  current  of  air  of  several  feet 
thickness  is  obtained  by  means  of  a  large  fan.  Various  and 
accurately  known  velocities  can  be  given  to  the  air  stream. 
The  surfaces  upon  which  we  wish  to  study  the  pressure  are, 
of  course,  of  limited  dimensions,  but  are  similar  to  those  which 

*  For  an  elementary  mathematical  treatment  see  Cowley  and  Evans,  "Aero- 
nautics in  Theory  and  Experiment,"  Longmans,  1918,  Chapter  III. 

f  Among  the  various  aerodynamical  laboratories,  can  be  mentioned  those 
of  M.  Eiffel  at  Paris,  the  National  Physical  Laboratory  of  England,  the  Massa- 
chusetts Institute  of  Technology,  and  Leland  Stanford  University.  A  description 
of  the  equipment  of  such  a  laboratory  can  be  found  in  Smithsonian  Miscellaneous 
Collection,  Vol.  62. 


THE  DYNAMICS    OF   THE   AIRPLANE 


are  to  be  employed  in  practice.  They  are  held  in  the  current 
of  air  by  arms  that  are  connected  with  balances  constructed 
so  as  to  allow  a  determination  of  the  magnitude  and  direction 
of  the  air  reaction.  By  changing  the  shape  of  the  surfaces, 
the  velocity  of  the  air  stream,  the  orientation  of  the  body,  etc., 
the  laws  which  furnish  the  basis  for  the  design  of  an  airplane, 
as  well  as  the  knowledge  of  its  behavior,  are  determined. 
Questions  relating  to  the  stability  of  a  machine,  to  the  proper- 
ties and  efficiency  of  propellers,  are  also  investigated  in  the 
same  way. 

We  pass  to  the  consideration  of  some  of  the  basic  aerodynamic 
laws,  as  determined  empirically. 

3.  Pressure  on  a  Plane. — When  a  portion  of  a  plane 
surface  is  introduced  normal  to  an  air  current,  the  pressure 
produced  upon  it  is  a  force  that  tends  to  displace  it,  and  we 
find  by  experiment  that  the  force  may  be  written 

F  =  KAV2, 

where  F  is  the  force,  A  the  area,  V  the  velocity  of  the  air 
stream,  and  K  a  constant  for  surfaces  geometrically  similar.* 
Strictly  speaking,  K  depends  upon  the  size  as  well  as  shape, 
but  changes  slowly  and  seems  to  approach  a  limiting  value  as 
the  plate  becomes  larger.  For  instance,  for  a  square  plate 
or  circle  we  find,  provided  A  is  expressed  in  square  feet,  V  in 
miles  per  hour,  F  in  pounds: 


Side  of  Square  or 
Diameter  of  Circle 

K 

in  Feet. 

o-5 

I.O 

.00269 
.00286 

2.0 

.00314 

3-o 

.00322 

5-o 

.00327 

IO.O 

.00327 

"This  law  can  also  be  obtained  from  theoretical  considerations.  Consider  the 
air  as  composed  of  separate  particles,  moving  parallel  and  striking  the  plate,  nor- 
mal to  their  direction  of  motion.  Assume  that  all  particles  are  brought  to  rest 


THE   PLANE   AND   CAMBERED   SURFACE  5 

We  note  the  slight  change  produced  in  K  when  the  side  of 
the  square  is  changed  from  3  to  10,  although  the  area  is  increased 
over  ten  times. 

The  value  of  K  will  depend  upon  the  units  in  which  we 
are  expressing  F,  A ,  and  V.  It  is  necessary  to  be  able  to  change 
from  one  system  to  another.  This  question  is  discussed  in  the 
appendix.  We  shall  assume,  unless  the  contrary  is  stated, 
that  forces  are  measured  in  pounds,  areas  in  square  feet,  and 
velocities  in  miles  per  hour. 

The  value  of  K  also  depends  upon  the  density  of  the  air. 
The  values  given  above  are  for  a  temperature  of  o°  C.,  and  a 
pressure  of  760  mm.  of  mercury. 

4.  The  Inclined  Plane. — Let  the  plane  be  inclined  at 
angle  0  to  the  direction  of  air  flow.  This  angle  is  called  the 
angle  of  attack.  Neglecting 
what  is  called  skin  friction,  it 
follows  that  the  resultant  force 
is  normal  to  the  plane.  Its 
magnitude  and  point  of  applica- 
tion vary  with  0.*  The  nature 
of  the  air  flow  about  the  plane  FlG  T 

is  quite  complicated,  but  can  be 

investigated  by  photography.  By  such  means  information 
is  obtained  as  to  how  the  air  divides  in  front  of  the  plane, 
flows  over  the  upper  and  lower  edges,  and  unites  again  behind 
the  plane. 

by  the  impact.  The  pressure  on  the  plate  will  equal  the  momentum  lost  by  the 
air.  The  quantity  of  air  that  comes  into  contact  with  the  plate  in  a  unit  of  time 
is  pA  V,  where  p  is  the  density  of  the  air.  As  all  particles  lose  their  velocity  V, 
the  momentum  lost  will  be  pA  V2.  The  errors  in  the  hypothesis  are  apparent,  but 
experiment,  while  giving  a  value  of  the  constant  different  from  that  deduced  by 
the  reasoning  above,  confirms  the  qualitative  nature  of  the  law. 

*  If  we  denote  by  Fe  the  value  of  the  force  for  the  angle  6  so  that  F^  repre- 
sents the  force  for  the  normal  plane,  various  formulae  exist  for  obtaining  FQ. 
Newton  gave  from  theoretical  reasons  F0  =  F90  sin2  6,  which  is  totally  discordant 
with  experiments.  The  formula  of  Colonel  Duchemin  is  much  more  accurate. 
He  gave 

2  sin  6 


THE   DYNAMICS   OF   THE   AIRPLANE 


We  are  especially  interested  in  the  vertical  and  horizontal 
components  of  the  total  force  F.  We  call  these  components 
the  Lift,  L,  and  Drag,  D.  It  is  proved  by  experiments  that 

we  may  write 

L  =  KVAV2,  D  =  KXAV2, 

where  Kv  and  Kx  are  constant  for  a  given  angle  of  attack. 
For  a  square  plane  we  can  take  the  following  values: 


Angle. 

Ky 

Kx 

5° 

.00045 

.00007 

10° 

.00097 

.OOOI9 

20° 

.00208 

.  00074 

30° 

.00291 

.00173 

5.  The  Center  of  Pressure. — It  is  important  to  know  the 
manner   in  which   the  point   of   application   of   the   resultant 

pressure   varies    as    the    angle    of 

attack    changes.     The  general  re- 
sult can  be  stated  as  follows: 

As  the  angle  of  attack  diminishes 
the  center  of  pressure  approaches 
the  leading  edge. 

This  behavior  of  the  center  of 
pressure  is  shown  in  Fig.  3. 

A  knowledge  of  the  movement 
of  the  center  of  pressure  is  of 
importance  in  the  subject  of  sta- 


30 


90 c 


FIG.  2. 


bility  and  in  finding  the  forces  on  the 
control  surfaces. 

6.  Aspect  Ratio. — It  was  stated 
above  that  the  quantity  K  in  the 
fundamental  equation  for  the  pressure 
was  constant  for  planes  geometrically 
similar.  In  case  we  have  a  rectangle 
of  sides  a  and  b,  situated,  as  shown, 
in  an  air  current,  we  call  the  fraction 


FIG.  3. 


a/b  the  aspect  ratio.     This  quantity  is  of  importance.     We 
find  that  the  coefficients  K,  Kv,  Kx,   which  have  been  used 


THE  PLANE   AND  CAMBERED   SURFACE 


above,  vary  when  the  aspect  ratio  is  changed.  Those  values 
which  have  been  given  for  a  square  may,  however,  be  used 
with  fairly  accurate  results  for  planes  with  aspect  ratio  close 
to  unity.  The  dependence  of  the  quantities  K,  KV)  Kx  on  the 
aspect  ratio  comes  from  the  important  effect  of  the  boundary 
of  the  surface  upon  the  resistance.  If  we  have  a  long  narrow 
rectangle,  it  is  evident  that  the  escape  of  the  air  around  the 
boundary  will  greatly  alter  the  pressure  from  what  it  would 
be  for  the  equivalent  square  plane. 

7.  The  Cambered  Wing. — If  we  examine  a  bird  wing,  we 
find  that  it  is  not  flat,  but  is  curved.  This  suggests  that  there 
may  be  some  aerodynamic  advantage  in  such  a  surface.  Experi- 
ment amply  confirms  this,  and  the  sustaining  surfaces  of  all 
airplanes  are  curved,  or  cambered.  Evidently  also  the  surface 
must  have  thickness  for  constructional  reasons. 

The  word  aerofoil  is  used  to  designate  a  sustaining  surface 
or  wing.  In  Fig.  4  there  is  illus- 
trated the  general  shape  of  the 
section  of  an  ordinary  aerofoil.  We 
call  AB  the  chord,  DC  the  camber 
of  the  upper  surface,  EC  the  camber 
of  the  lower  surface,  C  the  position  of  maximum  camber.  The 
quantities  BC,  CD,  CE  are  generally  expressed  in  terms  of 
the  chord  AB. 

Let  the  wing  be  placed  with  reference  to  the  air  flow  as 

shown  in  Fig.  5.  Then  by  the  angle 
of  attack  is  meant  the  angle  0  be- 
tween the  chord  and  the  direction 
of  the  relative  wind.  Let  N  repre- 
sent the  direction  of  the  normal  and 
R  the  resultant  pressure.  It  is  found 
that  for  small  angles  R  is  in  advance 
of  N.  This  increases  the  lift  L,  and 
diminishes  the  drag  D.  These  are 
desirable  effects,  and  it  is  partly  on  account  of  this  property  that 
the  cambered  wing  is  more  efficient  for  sustaining  purposes 
than  the  plane  wing. 


FIG.  4. 


FIG.  5. 


8 


THE   DYNAMICS   OF   THE   AIRPLANE 


It  is  evident  what  we  mean  by  the  terms  leading  edge, 
trailing  edge,  and  nose. 

8.  Characteristic  Curves  for  a  Given  Wing. — By  means 
of  experiments  in  a  wind  tunnel  we  investigate  R,  L,  D  as  func- 
tions of  V  and  6.  We  find  that  we  can  write,  as  for  a  flat  plane, 


=  KAV2, 


where 


=  KyAV2, 
=  KV2+KX2. 


=  KXAV2, 


The  quantities  K,  Kv,  and  Kx  are  again  constants  for  a  given 
angle  and  geometrically  similar  aerofoils.  The 'values  of  the 
coefficients  Kv  and  KX)  and  the  ratio  L/D  =  KV/KX  for  a  certain 
aerofoil  *  are  given  in  the  following  table.  The  table  also  gives 
the  position  of  the  center  of  pressure,  which  is  considered  in  §  12. 


Angle  of 
Atack. 

KV 

Kx 

LID 

Distance  of 
C.  P.  from 
Leading  Edge 

-4° 

—  .  000399 

.0001515 

2.64 

—  2 

+  .000156 

.0000905 

1.72 

—  I 

.000432 

.0000700 

6-15 

.620 

0 

.000721 

.  0000653 

11.00 

•530 

I 

.000936 

.0000670 

14.00 

•  463 

2 

.001146 

.  0000688 

16.60 

•  415 

4 

.001510 

.0000860 

17-50 

•  340 

6 

.001878 

.0001158 

16.  20 

.316 

8 

.002230 

.0001558 

14-30 

•303 

10 

.002580 

.0002055 

12.60 

.290 

12 

.002910 

.0002595 

ii.  20 

.283 

14 

.003165 

.  0003040 

10.40 

.274 

16 

.003165 

.0003710 

8.50 

.276 

18 

.  003080 

.0005520 

S-6o 

.310 

20 

.002882 

.0008500 

3-40 

.360 

In  order  to  use  these  values  to  determine  the  lift  and  drag, 
the  velocity  V  must  be  given  in  miles  per  hour,  the  area  A 
of  the  wing  in  square  feet.  The  values  of  L  and  D  that  are 
then  given  by  the  formula  will  be  in  pounds. 

*  This  wing  is  U.  S.  A.  No.  i  in  the  Third  Annual  Report  of  the  (American) 
Advisory  Committee  on  Aeronautics.  The  shape  of  the  wing  is  considered  in  §  10. 


THE  PLANE  AND  CAMBERED   SURFACE 


9 


The  values  of  Kv,  Kx  and  the  ratio  L/D  are  also  plotted 
in  the  following  curves.  It  is  to  be  noted  that  the  vertical 
scale  is  not  the  same  for  the  three  different  quantities. 


L 


X 


x 


V 


\ 


7 


A 


\ 


We  note  the  following  facts: 

i.  There  is  lift  at  an  angle  —2°,  i.e.,    sustentation  exists ] or  a 

negative  angle  of  attack. 


10 


THE   DYNAMICS    OF   THE   AIRPLA1 


2.  The  lift  increases  almost  as  a  linear  f 'unction  of  the  angle 
and  attains  a  maximum  at  about  15°,  then  decreases  rapidly. 

j.  The  drag  remains  sensibly  the  same  over  small  angles  and 
increases  very  abruptly  in  the  vicinity  0/15°. 

4.  The  ratio  L/D  increases  practically  as  a  linear  function  for 
small  angles,  and  attains  a  maximum  in  the  vicinity  0/15°. 

We  shall  merely  note  here  the  importance  of  the. ratio  L/D. 
For  horizontal  flight  the  lift  must  equal  the  weight  of  the 
machine.  Consequently  the  greater  the  quantity  L/D  the 


Ky 

.0030 
.0020 
.0010 

C 

x 

.-«  — 
14' 

-«i!> 

/ 

'12 

/ 

4 

4 

/ 

/6> 

/ 

/ 

r 

r 

r 

V 

*' 

\ 

2° 

.0002                         .0004         K, 

FIG.  7. 

less  resistance  there  is  to  be  overcome,  and  consequently  the 
less  power  is  necessary  for  a  given  speed  of  flight. 

It  is  from  a  study  of  the  characteristic  curves  for  a  given 
aerofoil  that  one  decides  upon  its  efficiency  or  suitability  for 
a  given  type  of  machine.  It  is  not  our  purpose  to  go  into 
this  question,  and  we  shall  merely  remark  that  the  type  of  wing 
to  be  selected  depends  upon  whether  the  machine  is  designed 
for  great  speed,  for  rapid  climbing,  or  for  carrying  heavy  loads. 

9.  Polar  Diagram. — Instead  of  plotting  the  lift  and  drag 
coefficients  with  the  angle  of  attack  as  argument,  we  can  plot 


THE  PLANE  AND  CAMBERED  SURFACE  11 

the  lift  against  the  drag.  It  gives,  however,  better  results  if 
if  we  take  different  scales  for  Ky  and  Kx.  We  obtain  in  this 
way  the  polar  diagram,  which  has  been  extensively  used  by 
M.  Eiffel.  In  what  follows  we  shall  see  its  suitability  for  many 
purposes. 

10.  The  section  of  any  wing  depends  upon  the  values  of 
BC,  DC,  EC,  measured  in  terms  of  the  chord  AB,  upon  the 
shape  of  the  nose,  and  the  general  shape  of  the  trailing  edge. 
Questions  of  strength  and  facility  of  construction  are  intimately 
connected  with  those  of  thickness.  By  varying  the  different 
elements  one  at  a  time,  we  are  able  to  arrive  at  conclusions 
as  to  the  best  value  of  any  element,  and  determine  the  best 
section  for  a  given  purpose.  As  an  average  we  can  state  that 
CB  equal  3/8,  and  CD  lies  between  0.05  and  0.08.  The  effect 
of  the  under  camber  is  not  so  well  known. 

In  Fig.  8  there  is  given  the  shape  of  the  wing  for  which 
the  curves  are  given  in  §  8. 


It  is  also  necessary  to  study  the  effect  of  the  shape  of  the 
ends  of  the  wing.  At  the  ends  leakage  occurs,  and  the  air 
flow  is  accordingly  greatly  modified.  A  wing  with  trailing 
edge  slightly  longer  than  leading  edge  is  found  to  be  most 
efficient. 

For  structural  reasons  the  aspect  ratio  does  not  usually 
exceed  8,  as  the  advantage  secured  from  increased  lift  is  then 
overbalanced  by  the  increased  weight  of  construction. 

11.  Pressure  over  the  Wing. — It  is  not  only  possible  to 
study  the  total  resultant  pressure  on  a  wing,  but  also  by  intro- 
ducing tubes  through  small  holes  in  the  surface,  and  con- 
necting them  to  a  manometer,  it  is  possible  to  determine  the 
pressure  at  different  points.  The  results  for  a  central  section 
are  shown  in  Fig.  9.  It  is  found  that  the  pressures  on  top 
of  the  wing  are  below  atmospheric.  Consequently  there  is 


12  THE  DYNAMICS   OF  THE  AIRPLANE 

suction  on  top  of  the  wing.  The  pressures  underneath  the 
wing  are  greater  than  atmospheric,  so  there  is  an  active  upward 
pressure.  The  two  combine  to  make  the  total  upward  pressure. 
In  the  figure,  the  suction  on  the  top  surface  is  represented  by 
lines  drawn  outward  from  it,  and  the  pressure  on  the  lower 
surface,  similarly  by  lines  drawn  outward.  It  is  seen  that 
the  suction  is  the  greater  of  the  two  forces.  In  fact,  it  con- 
tributes about  three-fourths  of  the  total  sustaining  force. 
This  also  shows  why  there  can  remain  sustentation  for  negative 
incidence. 


FIG.  9. 

We  find  also  in  this  way  that  we  must  not  make  the  chord 
too  great,  or  towards  the  trailing  edge  we  have  pressures  above, 
and  suction  below,  which  would  lessen  the  sustentation. 

In  the  same  way  the  pressures  along  any  section  can  be 
studied,  so  that  the  pressures  all  over  the  surface  can  be 
mapped,  and  we  obtain  a  very  clear  visualization  of  the  air 
reaction. 

It  should  be  noted  here  that  the  difference  of  pressures 
from  atmospheric  pressure  are  very  slight.  But  by  having 
large  wing  surfaces,  and  obtaining  sufficient  velocity  we  can 
make  the  total  lift  equal  to  or  greater  than  the  weight  of  the 
machine. 

12.  Center  of  Pressure. — The  resultant  pressure  is  a  vector, 
and  so  is  completely  fixed  in  magnitude  and  direction.  We 
know  all  there  is  to  know  about  it  if  we  know  its  magnitude 
and  its  moment  about  some  point,  for  instance  the  leading 
edge.  It  is,  however,  customary  to  speak  of  the  center  of 


THE  PLANE   AND   CAMBERED   SURFACE  13 

pressure,  which  we  define  as  the  point  where  the  vector  repre- 
senting the  pressure  intersects  the  chord. 

It  is  important  for  us  to  know  how  the  center  of  pressure 
behaves  as  the  angle  of  attack  changes.  In  general,  it  is  found 
for  cambered  surfaces  and  the  angles  employed  in  aviation 
that  the  following  is  true: 

The  center  of  pressure  recedes  from  the  edge  of  attack  as  the 
angle  of  attack  diminishes.  For  larger  and  increasing  angles 
it  recedes.  This  property  should  be  contrasted  with  that  given 
above  for  plane  surfaces. 

Fig.  10  shows  the  motion  of  the  center  of  pressure  for  the 
wing  considered  in  §  8. 


FIG.  10. 

13.  Relation  between  Ky  and  Kx.  —  It  is  impossible  to 
obtain  a  simple  relation  between  the  coefficients  Ky  and  Kx, 
but  an  approximate  one  will  be  obtained,  and  use  will  be 
made  of  it  later. 

In  the  note  to  §  4  there  is  given  the  formula  of  Colonel 
Duchemin  for  the  pressure  on  a  flat  plane  as  a  function  of  the 
angle  of  attack.  It  is  obvious  that  for  small  angles  it  varies 
approximately  as  the  sine  of  the  angle.  Let  us  assume  such  a 
relation  for  a  cambered  wing.  As  there  is  still  lift  for  a  negative 
incidence  we  would  expect  to  measure  angles  from  the  position 
of  the  wing  that  gives  no  lift.  We  shall  not  do  this,  however, 
and  the  results  obtained  will  not  be  valid  for  the  small  angles 
of  attack. 

We  assume  a  relation 


where  K  is  the  coefficient  of  pressure,   and  K'  a  constant. 


14  THE  DYNAMICS  OF  THE  AIRPLANE 

Assuming  that  the  pressure  is  normal  to  the  chord  we  would 
then  have 

Ky  =  K'  sin  e  cos  6,         Kx  =  K'  sin2  0. 
Hence 


Kv2K'cos26' 

If  6  is  small  this  is  approximately  constant. 

In  order  to  see  something  about  the  accuracy  of  our  result 
we  shall  actually  calculate  the  value  of  Kx/Kj2  for  the  values 
given  in  §  8.  The  results  are  as  follows: 

Angle       i°     2°     4°     6°     8°     10°    12°    14° 

K*/Kf       76.40     52.3     47.4    33.6    31.3     30.8    30.6    30.3 

While  the  value  is  not  constant,  it  is  seen  that  between  6°  and  14° 
it  is  practically  so,  and  approximate  results  can  be  obtained  by 
assuming  that  it  is  constant. 

14.  The  Biplane.  —  In  order  to  secure  greater  lifting  forces, 
and  at  the  same  time  deal  easily  with  the  problem  of  con- 

struction,  it  is  customary  to 
use  two  or  more  surfaces,  one 
above  another.  We  shall  limit 
ourselves  to  the  biplane. 

By  the  gap  is  meant  the 
distance  between  the  two 
planes,    measured   in  terms 
of   the   chord,   i.e.,   the    ratio 
•        BC/AB.     By   the   stagger   is 

meant   the  distance  CD,  also 
FIG.  ii.  ,    .  r      .  _ 

measured    in    terms    of    AB. 

The  stagger  is  positive  if  the  upper  plane  is  in  advance  of  the 
lower,  negative  if  in  the  rear. 

The  question  of  the  relative  efficiency  of  the  upper  and  lower 
wings  enters  at  once.  The  gap  is  limited  by  constructional 
reasons,  and  usually  varies  between  i  and  i.i.  It  is  found 
then  that  the  efficiency  of  the  lower  wing  is  considerably  lessened 
by  the  presence  of  the  upper  wing.  The  reason  becomes 
apparent  when  we  recall  what  was  said  in  §  u.  The  region  of 


THE  PLANE   AND   CAMBERED   SURFACE  15 

depression  above  the  lower  wing  is  considerably  modified  by 
the  upper  aerofoil.  And  as  this  depression  is  what  gives  the 
preponderant  part  of  the  lift  we  would  expect  the  lift  to  be 
lessened.  On  the  other  hand  the  suction  above  the  upper 
wing  is  unimpaired,  while  the  less  important  pressure  below  is 
somewhat  modified.  The  lower  wing  then  has  less  efficiency 
than  the  upper  one,  and  on  this  account  its  aspect  ratio  is 
sometimes  made  smaller  than  that  of  the  upper  wing. 

The  center  of  pressure  for  a  biplane  is  purely  a  matter  of 
definition.  We  shall  understand  it  to  mean  the  point  of  inter- 
section of  the  vector  representing  the  pressure  with  a  line  parallel 
to  the  chords  and  midway  between  them. 

15.  Body   Resistance. — In    the   forward   movement   of   an 
airplane,  all  parts  of  the  machine  contribute  to  the  resistance 
that  must  be  overcome,  while  the  aerofoils  alone  contribute  to 
the  sustentation.*    It  is  necessary  to  make  the  resistance  of 
all  parts  of  the  machine  as  small  as  possible.     This  is  done  by 
giving  proper  shapes  to  fuselage,  struts,  wires,  landing  gear, 
etc.     The  resistance  that  arises  from  the  parts  other  than  the 
wings  can    be  conceived  of  as  arising  from  the  motion  of  a 
square  plane  normal  to  the  direction  of  motion.     We  call  this 
the  equivalent  detrimental  surface.      Let  its  area  be  s;    the 
resistance  due  to  the  body  can  then  be  written 

f=ksV2. 

For  the  complete  machine  we  therefore  have 
L  =  KyA  V2,  D  =  KXA  V2+ksV2. 

16.  Experiments  on  Complete  Model. — We  have  thus  far 
discussed    experiments    made    on    the    models    of    the    wings. 
Models  of  complete  machines,  with  the  exception  of  propeller, 
wires,  etc.,  whose  contribution  to  the  complete  resistance  is 
small,  are  also  subjected  to  exhaustive  study  in  the  wind  tunnel. 
Data  necessary  for  the  discussion  of  stability  are  obtained  in 

*  The  body  doubtless  adds  something  to  the  sustentation,  but  it  is  not  an 
amount  of  which  we  can  take  account,  and  is  negligible  when  compared  with  that 
furnished  by  the  wings. 


16 


THE   DYNAMICS   OF   THE   AIRPLANE 


THE  PLANE  AND  CAMBERED   SURFACE  17 

this  way.  A  diagram  of  a  model  of  this  sort  is  shown  in  Fig.  12.* 
Lines  showing  the  direction  of  the  air  reaction  for  angles  of 
attack  varying  from  — 1°  to  8°  are  shown. 

A  question  of  primary  importance  at  once  arises.  To 
what  extent,  and  under  what  conditions  are  the  results  obtained 
from  experiments  on  models  applicable  to  full-scale  machines? 
The  study  of  this  question  depends  upon  what  is  called  dynamical 
similarity.  We  shall  not  touch  upon  it  here.  In  the  appendix 
reference  will  be  given  to  treatments  of  this  important  question. 

*  This  is  taken  from  Hunsaker's  paper,  "Dynamical  Stability  of  Aeroplanes," 
Smithsonian  Miscellaneous  Collections,  Vol.  62.  No.  5,  1916.  It  is  a  model  of  a 
biplane  tractor  designed  by  Captain  V.  E.  Clark,  U.  S.  A.,  and  is  representative 
of  modern  design. 


CHAPTER  II 
STRAIGHT  HORIZONTAL  FLIGHT 

17.  LET  us  consider  the  motion  of  an  airplane.  Suppose 
the  motor  is  running.  The  machine  is  at  any  instant  acted 
upon  by  three  forces,  its  weight,  acting  downwards  through 
the  center  of  gravity,  the  traction  of  the  propeller,  and  the 
resistance  of  the  air.  As  the  machine  is  a  rigid  body  we  can 
in  general  combine  these  three  forces  into  a  single  force  and  a 
couple.  The  force  determines  the  instantaneous  motion  of  the 
center  of  gravity,  and  the  couple  determines  the  rotation  which 
the  machine  is  momentarily  undergoing.  When  regarded  in 
this  general  way  the  problem  is  seen  to  be  very  complex,  on 
account  of  the  nature  of  the  resistance  of  the  air,  for  this 
resistance  depends  upon  the  angle  at  which  the  wings  are  any 
instant  attacking  the  air,  and  on  the  velocity.  Thus  both  the 
motion  of  the  center  of  gravity  and  the  rotational  motion 
affect  it. 

In  order  to  fix  the  ideas,  suppose  the  airplane  moves  con- 
stantly in  a  vertical  plane,  coincident  with  the  plane  of  sym- 
metry of  the  machine;  then  the  center  of  gravity  traces  out 
a  certain  plane  trajectory.  Suppose  further  that  the  angle 
of  attack  remains  constant;  the  angle  between  the  tangent 
to  the  trajectory  and  the  chord  of  the  wings  is  then  constant. 
The  relative  .wind  at  any  instant  is  in  the  direction  of  the 
tangent.  Furthermore,  it  is  evident  that  what  we  have  called 
the  lift  in  the  preceding  chapter  will  now  be  in  the  direction 
of  the  normal  to  the  trajectory,  and  the  drag  will  be  in  the 
direction  of  the  tangent.  The  motion  of  the  machine  will  then 
be  determined  by  a  force  equal  to  the  weight  downward  through 

18 


STRAIGHT  HORIZONTAL  FLIGHT  19 

the  center  of  gravity,  the  traction  of  the  propeller  along  the 
tangent  to  the  path,  the  lift 


2 


KVA 

9 

perpendicular  to  the  tangent,  and  the  combined  drag  of  the  wings 
and  body 


along  the  tangent  to  the  path. 

18.  Horizontal  Flight.  —  We  consider  first  the  simplest 
possible  case.  Let  the  machine  be  moving  horizontally  in  a 
straight  line,  with  constant  angle  of  attack.  What  are  the 
conditions  that  must  be  fulfilled  F 
to  make  this  possible,  and  what 
properties  about  the  motion  can  be 
discovered? 

The  lift  L  is  now  a  vertical  force, 
and  the  drag  a  horizontal  one.  In 
the  figure,  F  represents  the  resultant 
air  pressure,  applied  at  P,  W  repre- 
sents the  weight  applied  at  the 
center  of  gravity,  and  T  the  traction 
of  the  propeller  applied  at  a  point 

B,   any  point  in   the   axis   of   the 

„  FIG.  13. 

propeller. 

We  are  supposing  that  the  machine  is  moving  horizontally. 
Hence  W  =  L.  Furthermore,  we  must  have  T  =  D,  for  if,  for 
instance,  T>D,  the  velocity  would  start  to  increase.  This 
would  increase  L,  and  the  machine  would  start  to  rise.  Like- 
wise, if  T<D,  the  machine  would  start  downwards.  Hence, 
at  a  constant  angle  of  attack,  uniform  motion  alone  is  possible, 
if  the  machine  is  to  fly  horizontally. 

In  order  that  there  be  no  rotation,  the  couple  formed  by 
W  and  L  must  be  equal  to  that  formed  by  D  and  T,  and  have 
the  opposite  sense.  Consequently  the  three  forces,  F,  T,  and 
W  must  be  concurrent. 


20  THE   DYNAMICS   OF   THE   AIRPLANE 

The  equations  of  motion  are  therefore: 
W  =  KVAV2, 
T  =  KxAV2+ksV2. 

19.  The  Velocity.— The  first  of  these  equations  is  called  the 
equation  of  sustentation.     From  it  we  find  the  velocity 


with  which  the  machine  must  move  to  produce  sustentation. 
From  the  manner  in  which  Kv  varies  with  the  angle  of  attack, 
we  see  that  the  smaller  the  angle  of  attack  is,  the  greater  must 
be  the  velocity.*  Furthermore,  the  velocity  increases  with 
the  quantity  W/A.  This  quantity  is  the  weight  per  unit 
area,  or  the  loading,  and  we  see  the  importance  that  it  plays. 
It  is  not  the  weight,  but  the  loading,  that  determines  the  speed 
of  the  plane  for  a  given  angle  of  attack. 

20.  Landing  Speed. — It  is  apparent  that  landing  must  be 
accomplished  by  flying  horizontally  very  close  to  the  ground 
at  the  smallest  velocity  possible.  This  will  be  with  the  angle 
of  attack  giving  the  greatest  value  of  Kv.  If  we  have  chosen 
the  type  of  wing,  and  the  weight  of  the  machine,  we  can  obtain 
the  area  of  the  wing  by  fixing  the  landing  speed. 

Suppose  we  wish  to  construct  a  monoplane  weighing  1200 
pounds,  the  curves  for  the  wing  being  those  in  §  8.  Let  the 
landing  speed  be  45  miles  per  hour.  The  maximum  lift  coeffi- 
cient available  we  take  as  0.00316,  corresponding  to  an  angle 
of  attack  of  about  14°.  We  have  then 

187.5  square  feet. 


.00316X45' 

This  gives  us  a  loading  of  6.40  pounds  per  square  foot. 

The  value  of  the  velocity  of  the  machine  for  different  angles 
of  attack  is  best  visualized  by  a  curve.     Using  the  loading 

The  means  employed  to  cause   the  machine   to   fly  at   different  angles   of 
attack  is  considered  in  Chap.  VII,  where  controllability  is  discussed.     See  also  §  30. 


STRAIGHT  HORIZONTAL  FLIGHT 


21 


that  we  have  just  found  and  the  values  of  Ky  from  §  8  we 
obtain  the  curve  given  in  Fig.  14. 

21.  Effect  of  Altitude. — With  the  same  angle  of  attack  a 
machine  will  not  fly  horizontally  at  different  altitudes  with  the 
same  velocity,  on  account  of  the  change  in  the  density  of  the 
air  as  we  ascend.  The  coefficients  Kv,  Kx  and  k  all  vary 
directly  as  the  density  of  the  air.  Thus,  if  we  let  Kv(z),  Kx(z), 
k(z)  represent  the  values  at  altitude  z,  and  Kv,  Kx,  k  the  values 


140 


100 


20! 


-2° 


7°  10° 

Angle  of  Attack 

FIG.  14. 


13° 


16' 


at  the  surface  of  the  earth,  which  are  the  values  actually  given 
for  any  wing,  we  have 


where  p(z)  denotes  the  ratio  of  the  density  of  the  air  at  the 
altitude  z  to  the  density  at  the  surface  of  the  earth.  The 
value  of  the  decreasing  function  p(z)  is  given  in  a  table  in  the 
Appendix. 

It  is  evident  that  for  a  given  angle  of  attack  the  velocity 
of  the  machine  varies  inversely  as  the  square  root  of  the  density 


22  THE  DYNAMICS   OF  THE   AIRPLANE 

of  the  air.     For  a  given  angle  of  attack  the  machine  therefore 
flies  faster  as  the  altitude  increases.* 

22.  The  Effort  of  Traction.  —  The  traction  that  the  propeller 
must  furnish  is  given  by  the  second  equation  in  §  18: 


If  we  take  the  quotient  of  T  and  W  we  obtain 


W  Ky 

It  is  evident  that  the  right-hand  side  is  independent  of  the 
altitude,  for  Ky,  Kx,  k  are  all  multiplied  by  the  same  quantity 
as  the  altitude  increases.  The  traction,  therefore,  does  not  depend 
upon  the  altitude,  but  only  on  the  angle  of  the  attack. 

23.  Optimum  Angle.  —  Consider  the  polar  curve  given  in 
§  9.  Suppose  for  a  moment  that  the  unit  used  along  the  two 
axes  is  the  same.  Take  a  point  Q  at  distance  ks/A  to  the  left 
of  the  origin.  Let  P  be  a  point  on  the  curve  corresponding  to 
an  -angle  of  attack  a.  Then  it  is  evident  that 

- 

where  <£  is  the  angle  that  QP  makes  with  the  Ky-a,xis.  It  is 
seen  that  QP  will,  in  general,  intersect  the  polar  in  another 
point  P',  corresponding  to  a  different  angle  of  attack,  a  '.  But 
the  speeds  necessary  for  sustentation  at  angles  of  attack  a 
and  a  are  different.  We  can  therefore  fly  at  two  different 
speeds,  and  experience  the  same  resistance,  and  thus  require 
the  same  traction  from  the  propeller. 

The  effort  of  traction  will  be  a  minimum  when  tan  0  is  a 
minimum,  and  this  occurs  when  QP  is  tangent  to  the  polar. 
Let  Mi  be  the  point  of  tangency,  and  a\  be  the  corresponding 
angle  of  attack.  The  angle  «i  is  called  the  optimum  angle. 

*  We  really  should  say,  the  machine  must  be  made  to  fly  faster  in  order  to 
produce  sustentation.  What  requirement  this  puts  on  the  power  that  the 
motor  is  furnishing  will  be  considered  later. 


STRAIGHT   HORIZONTAL   FLIGHT 


23 


In  case  we  have  used  different  horizontal  and  vertical 
scales  on  the  polar,  as  is  convenient,  we  can  no  longer  obtain 
T/W  by  merely  taking  tan  <£.  However,  we  shall  still  obtain 
ai  by  taking  the  point  Mi  such  that  QM\  is  tangent  to  the 
polar.  We  can  also  find  the  second  angle  of  attack  for  which 
the  traction  is  the  same  as  that  for  some  given  angle,  by  finding 
as  before,  the  second  point  of  intersection  of  QP  and  the  polar. 
The  distance  OQ  must  be  plotted  in  the  same  unit  used  for  Kx. 


FIG.  15. 

24.  Fineness. — The  value  of  the  ratio  T/W  is  called  the 
fineness  for  a  given  angle  of  attack.  It  is  easily  calculated 
for  a  machine  when  we  know  Kv,  Kx,  s,  and  A .  The  importance 
that  it  plays  in  the  performance  of  the  machine  will  appear 
as  we  proceed.  In  certain  respects  the  fineness  completely 
characterizes  the  machine.  It  is  desirable  that  the  fineness 
be  as  small  as  possible.  For  a  given  value  of  the  fineness 
the  traction  necessary  for  sustentation  varies  directly  as  the 
weight  of  the  machine.  The  minimum  value  of  T/W  is  some- 
times spoken  of  as  the  fineness  of  the  machine. 

Let  us  assume  a  detrimental  surface  of  8  square  feet,  and 
take  A  =  187.5,  as  found  in  §  20.  From  §  3  we  have  £  =  .00322. 
This  gives  ks/A  =.0001373.  The  value  of  the  fineness  for  the 
airplane  considered  is  then  found  to  be  that  given  in  the 
following  table.  We  denote  it  by  the  letter  B.  The  traction 
T  is  also  given,  for  the  weight  W  =  1200. 


24 


THE   DYNAMICS   OF    THE   AIRPLANE 


Angle  of 
Attack. 

Fineness 
B. 

Traction  T, 
in  Ibs. 

To 

J. 

.604 

729 

O 

.281 

337 

I 

.218 

261 

2 

.179 

216 

4 

-i47 

176 

6 

•135 

161 

8 

•131 

157 

10 

•  133 

159 

12 

.136 

163 

14 

•139 

167 

16 

.  160 

192 

From  the  results  given  in  the  table  it  is  possible  to  plot 
the  traction  against  the  angle  of  attack.  From  the  curve 
which  is  derived  and  the  velocity  curve  in  Fig.  14,  it  is  then 
possible  to  plot  the  traction  against  the  velocity.  Such  a 
curve  will  be  given  in  the  same  diagram  with  the  power,  Fig.  16. 

25.  It  is  useful  at  times  to  consider  separately  the  resist- 
ances that  arise  from  the  wings  and  the  body  of  the  machine, 
respectively.  It  is  apparent  that  curves  representing  these 
quantities  will  be  obtained  by  plotting  KxAV2  and  ksV2  against 
the  velocity. 

An  application  of  the  work  of  §  13  can  be  made  in  con- 
nection with  the  determination  of  the  optimum  angle.  We 
desire  the  minimum  of  the  quantity 


considered  as  a  function  of  the  angle  of  attack.     The  product 
of  the  two  terms  is 

ksAKxV4. 

Inserting  the  value  of  V  derived  from  the  equation  of  susten- 
tation  this  becomes 

ksW2  Kx 
A     Kf 


STRAIGHT   HORIZONTAL   FLIGHT  25 

The  coefficient  of  Kx/K2y  is  a  constant,  and  from  §  13  it 
follows  that  Kx/Ky2  itself  is  practically  constant  in  the  vicinity 
of  the  optimum  angle.  Now  from  the  principles  of  maxima 
and  minima  we  know  that  the  sum  of  two  quantities  whose 
product  is  constant  attains  its  least  value  when  the  two  quan- 
tities are  equal.  Accordingly,  it  follows,  with  a  sufficient 
approximation,  that  the  optimum  angle  is  the  angle  for  which 
the  drag  of  the  wings  equals  that  of  the  body  of  the  machine. 
Therefore  if  we  plot  the  drag  of  the  wings  and  of  the  machine 
separately  against  the  velocity,  the  optimum  angle  will  be  the 
angle  that  corresponds  to  the  point  where  the  two  curves  cross, 
and  the  velocity  for  the  optimum  angle  can  be  obtained  at 
once. 

26.  The  Useful  Power. — The  useful  power  developed  by 
the  motor  is  TV,  which  can  be  written 

P  =  KzAV3+ksV3 

£_  Kx+ks/A 

y*~  Ky  ' 

if  we  desire  to  put  the  fineness  in  evidence. 

The  curve  of  the  useful  power  necessary  is  easily  obtained 
from  that  of  the  traction  by  multiplying  the  ordinate  by  the 
abscissa.*  The  traction  and  useful  power,  plotted  against 
velocity,  are  given  in  Fig.  16. 

27.  In  addition  to  overcoming  the  resistance  and  imparting 
the  velocity  necessary  for  sustentation,  the  propeller  imparts 
a  velocity  to  the  air  with  which  it  comes  in  contact.     The 
force  that  is  expended  in  this  way  is  equal  to  the  momentum 
imparted  to  the  air.     The  resultant  energy  is  lost.     The  only 
part  of  the  total  work  the  motor  is  developing  that  is  useful 
to  us  is  what  is  considered  above.     We  can  calculate  from  the 
characteristic  curves  of  the  machine  the  useful  power  which 

*  To  obtain  the  power  we  may  express  V  in  feet  per  second.  The  product  TV 
then  gives  the  power  in  foot-pound-seconds.  To  change  to  horsepower  divide 
by  550.  If  V  is  in  miles  per  hour  and  T  in  pounds  we  obtain  horsepower  by 
dividing  the  product  TV  by  375. 


26 


THE   DYNAMICS   OF   THE   AIRPLANE 


we  must  have  for  each  angle  of  attack.  The  total  power  which 
the  engine  must  be  furnishing  is  then  dependent  on  the  efficiency 
of  the  propeller.  This  question  will  be  considered  in  its  proper 
place. 

28.  The  Economical  Angle. — Just  as  there  is  an  angle  of 
attack  for  which  the  effort  of  traction  is  a  minimum,  so  there 


100        400 


•60   o 

&  200 
40 


100 


20 


40 


50 


70  80 

Velocity,  Mi.  hr. 

FIG.  1 6. 


100 


is  an  angle  for  which  the  power  necessary  is  the  least  possible. 
This  angle  is  not  the  same  as  the  optimum  angle.  It  is  evident 
at  once  that  it  must  then  be  greater.  For  consider  the  product 
TV.  As  the  angle  of  attack  decreases  from  the  optimum 
angle  a\  both  T  and  V  increase.  Hence  P  increases  con- 
tinuously and  could  not  attain  a  minimum.  As  the  angle  in- 
creases from  OLI  the  traction  T  increases  and  V  decreases,  so 
the  product  TV  may  have  a  minimum.  Furthermore,  we  see 
that  in  order  that  the  product  TV  should  have  a  minimum 
coincidently  with  T}  it  is  necessary  that  either  V  have  a  minimum 
at  the  same  time,  or  that  T  vanish  at  its  minimum,  neither  of 


STRAIGHT   HORIZONTAL   FLIGHT 


27 


which  conditions  is  fulfilled.  Hence  the  power  will  reach  a 
minimum  at  an  angle  of  attack  greater  than  the  optimum 
angle. 

The  angle  for  minimum  power  can  be  found  from  the  true 
polar,  that  is,  the  polar  constructed  with  equal  units  along 
the  two  axes.  Take  Q  as  origin  and  let  x  and  y  be  the  Cartesian 
coordinates  of  a  point  on  the  polar.  Then  from  §  24  the  power 
can  be  written 


where  h  is  constant.     To  find  the  minimum  of  this  we  equate 
its  derivative  to  zero,  giving 

dx  =  $x 

dy     2y' 


FIG.  17. 

Suppose  the  tangent  at  any  point  of  the  polar  curve  makes  an 
angle  ^  with  the  y  axis.  Then  the  condition  just  found  gives 
for  the  point  of  minimum  power 

tan  ^  =  f  tan  </>, 

where  </>  has  the  meaning  given  to  it  in  §  23.  At  the  optimum 
angle  we  have  ^  =  0  and  from  the  shape  of  the  polar  curve 
we  see  that  the  point  M2  satisfying  the  last  condition  is  to 


28  THE   DYNAMICS   OF   THP;  AIRPLANE 

the  right  of  Mi,  the  point  corresponding  to  the  optimum  angle. 
The  corresponding  angle  of  attack,  which  we  shall  denote  by  a2, 
is  called  the  economical  angle.  Flight  at  that  angle  requires 
a  minimum  rate  of  consumption  of  fuel,  assuming  that  the 
efficiency  of  the  propeller  is  practically  constant  over  the 
region  considered. 

It  is  not  difficult  to  show  that  the  equation  last  written 
will  also  be  true  at  the  points  M2  and  Mi  if  the  polar  is  con- 
structed with  different  scales  along  Kv  and  Kx. 

We  can  summarize  the  results  of  this  section  as  follows: 

The  economical  angle  is  greater  than  the  optimum  angle. 

The  machine  flies  faster  at  the  optimum  angle  than  at  the 
economical  angle. 

29.  Although  the  consumption  of  fuel  is  at  a  less  rate  at 
the  economical  angle,  the  machine  flies  more  slowly,  so  con- 
sumption takes  place  over  a  longer  period,  in  case  a  given 
distance  is  to  be  traversed.  In  fact  the  work  per  unit  distance 
traversed  is  exactly  equal  to  the  traction,  and  is  therefore  a 
minimum  when  the  traction  is  least.  Furthermore,  high  speed 
is  usually  a  desirable  feature,  so  flight  at  the  optimum  angle 
is  preferable  to  flight  at  the  economical  angle. 

We  can  obtain  some  interesting  results  by  comparing  the 
increased  speed  with  the  increased  power  necessary  when 
flying  at  an  angle  other  than  the  economical  one. 

Let  V2j  PI  represent,  respectively,  the  velocity  and  power 
for  the  economical  angle,  point  M2  on  the  polar,  Fig.  17,  V 
and  P  the  corresponding  quantities  for  any  other  angle,  repre- 
sented by  M  on  the  polar,  Ky,2  and  Kv,  the  corresponding  lift 
coefficients.  Then 


V2      v  Kv'        P2         #,   tan0f' 

where  <£2  and  <£  are  the  angles  in  the  polar  diagram  referring  to 
the  conditions  of  flight.     We  can  then  write 

V      P    (tan  02 


V2    P2      tan 


STRAIGHT   HORIZONTAL   FLIGHT  29 

Let  Mz  be  the  other  point  where  OMi  intersects  the  polar,  and 
«/  the  corresponding  angle  of  attack.  Then  if  M  lies  between 
Mz  and  Mz  the  quantity  tan  <£2/tan  0  is  greater  than  unity. 
Consequently,  for  angles  of  attack  between  a,  and  «/  the  speed 
increases  at  a  greater  rate  than  the  power  necessary.  It  is  not 
until  the  angle  of  attack  falls  below  «/  that  the  increased 
speed  is  obtained  by  a  disproportionate  increase  in  necessary 
power,  and  therefore  fuel.*  We  see  also  that  the  farther  M<z 
is  from  Mi,  the  point  representing  the  optimum  angle,  the 
greater  will  be  the  range  in  values  for  the  angle  of  attack,  for 
which  the  increase  of  speed  is  not  accompanied  by  a  too 
great  increase  of  power. 

30.  An  application  of  the  results  that  have  been  obtained 
can  be  made  to  the  question  of  control  of  the  machine.  For 
a  given  angle  of  attack  the  machine  can  fly  horizontally  with 
but  one  velocity.  Alteration  in  velocity  must  therefore  be 
secured  by  changing  the  angle  of  attack.  This  is  done  by  means 
of  the  elevator,  whose  effect,  as  explained  in  §  94,  is  to  rotate 
the  machine  as  a  whole  about  an  axis  perpendicular  to  the 
plane  of  symmetry.  It  is  apparent  that  there  is  a  single  angle 
of  attack  for  which  the  axis  of  the  propeller  is  horizontal. 
But  the  vertical  component  of  the  propeller  thrust  is  in  all 
instances  small,  and  it  has  been  neglected.  As  the  power  neces- 
sary varies  with  the  speed  (or  angle  of  at  tack),  a  manipulation 
of  the  elevator  for  the  purpose  of  altering  the  speed  must 
be  accompanied  by  a  change  in  the  admission  of  gasoline  to  the 
motor.  Otherwise  the  machine  would  start  to  ascend  or  descend. 

We  have  considered  only  the  useful  power  necessary  to 
maintain  flight  at  different  velocities.  The  total  power  the 
motor  is  developing  is  determined  by  the  number  of  revolutions 
it  is  making  per  minute.  The  quotient  of  the  useful  power 
and  the  total  power  being  developed  by  the  motor  is  the  efficiency 
of  the  propeller.  This  is  not  constant,  but  depends  upon  both 
forward  velocity  and  number  of  revolutions  per  minute.  The 
question  is  discussed  more  at  length  in  Chapter  V.  It  is 

*  The  statement  about  the  increase  of  fuel  is  only  approximately  correct,  as 
the  efficiency  of  the  propeller  must  be  considered. 


30  THE   DYNAMICS   OF   THE   AIRPLANE 

apparent,  however,  from  what  has  been  said,  that  as  the  pilot 
simultaneously  changes  the  elevator  and  the  admission  of 
gasoline  in  order  to  secure  a  different  horizontal  velocity,  the 
efficiency  of  the  propeller  may  be  either  increasing  or  decreasing. 
We  shall,  however,  assume  that  an  increase  of  useful  power 
will  call  for  an  increase  of  total  power,  and  therefore  require 
an  increase  in  the  admission  of  gasoline.  Conversely,  we 
shall  assume  that  a  decrease  of  useful  power  will  be  secured 
through  a  decrease  in  the  amount  of  gasoline  admitted  to  the 
motor. 

Suppose  the  machine  is  flying  at  an  angle  of  attack  less 
than  the  economical  angle.  To  fly  faster  the  pilot  must,  as 
he  properly  manipulates  the  elevator,  admit  more  gasoline  to 
the  motor.  Conversely,  if  he  desires  to  fly  slower,  he  must 
cut  down  on  the  admission  of  gasoline.  If,  however,  the 
machine  is  flying  at  an  angle  of  attack  greater  than  the  econom- 
ical angle,  there  is  a  reversal  in  the  use  of  the  controls.  Greater 
speed  will  be  secured  by  lessening  the  admission  of  gasoline, 
as  the  elevator  is  turned  so  as  to  decrease  the  angle  of  attack, 
while  more  gasoline  must  be  admitted  if  it  is  desired  to  fly 
slower.  If  the  machine  is  flying  at  the  economical  angle,  it 
will  require  more  gasoline  to  fly  either  faster  or  slower. 


CHAPTER  III 


DESCENT  AND   ASCENT 

i.  DESCENT 

31.  In  the  last  chapter  we  have  discussed  the  conditions 
of   rectilinear   horizontal  flight,  considering  such   questions  as 
velocity,  traction,  and  power.     The  results  form  the  basis  for 
many  problems  connected  with  design  and  performance  of  a 
machine.     Other  conditions  of  flight  must,  however,  be  con- 
sidered, and  we  shall  first  take  up  that  of  descent.     Descent, 
of  course,  can  be  accomplished  in  numerous  ways  by  the  proper 
manipulation    of    the    controls.     There    are,    however,    certain 
conditions  of  descent  of  a  more  definite  character,  which  give 
rise  to  some  interesting  questions. 

32.  Equations    of    Motion. — Consider    a   part   of   the    tra- 
jectory, starting  at  a  point  P.    Let  G  be  the  position  of  the 
center  of  gravity  at  a  certain 

instant,  and  0  the  angle  that 
the  tangent  makes  with  the 
horizontal.  Since  the  lift  and 
drag  are  normal  to  and  along 
the  tangent,  respectively,  we 
have  for  the  total  forces  in  these 
two  directions, 

WcosB-Ky(z)AV2, 
T+Wsm  8-(Kx(z)A+k(z)s)V2 

where  T  is   the  traction,  and  z 
is  the  altitude  of  the  machine. 
The  dependence  of  the  various  coefficients  on  the  air  density 
has  been  put  in  evidence  because  we  shall  no  longer  be  con- 

31 


FIG.  1 8. 


32  THE   DYNAMICS   OF   THE   AIRPLANE 

sidering  the  machine  at  a  constant  altitude.  We  are  making 
the  assumption  that  the  axis  of  the  machine  will  maintain 
itself  along  the  tangent,  so  that  the  angle  of  attack  remains 
constant.  If  the  trajectory  is  curved  this  will  necessitate  a 
rotation  of  the  machine  about  an  axis  perpendicular  to  the  plane 
of  symmetry.  The  stabilizing  surfaces  of  the  machine  provide 
for  such  a  behavior. 

From  analytical  mechanics  we  know  that  the  normal  acceler- 
ation of  a  particle  is  given  by  V2/r,  where  r  is  the  radius  of 
curvature  of  the  path,  while  the  tangential  acceleration  is 

d2s 

given  by  —  -,  where  s  is  the  length  of  arc  measured  from  a 
dt 

convenient  point.     But  we  can  write 

F2=       de     d2s_dV_dVds_vdV^idV2 
r  ~        ds      dt2~  dt  ~  ds  dt~      ds  ~  2   ds  ' 

Since  the  velocity  enters  uniformly  to  the  second  power  in 
all  expressions  under  consideration  we  shall  make  the  change 
of  variables  V2  =  u.  Before  writing  the  equations  of  motion 
we  shall  make  another  simplification.  As  the  machine  descends 
it  encounters  air  of  increasing  density.  Therefore  Kv(z), 
Kx(z),  k(z)  are  all  increasing  functions.  We  write 


where  the  function  /(s)  is  an  increasing  function  of  s,  assuming 
that  the  machine  constantly  descends.  (The  fact  that  s  has 
also  been  used  to  represent  the  detrimental  surface  will  evi- 
dently cause  no  trouble,  as  it  will  no  longer  appear  in  that 
significance  on  account  of  the  substitutions  above.)  We  shall 
measure  arc  from  the  point  P.  We  thus  have/(o)  =  i. 
The  equations  of  motion  are  therefore: 

mu  —  =  W  cos  0  —  af(s)u, 
as 


2  ds 
where  m  represents  the  mass  of  the  machine. 


DESCENT  AND  ASCENT  33 

33.  Rectilinear  Descent.  —  We  shall  now  investigate  the 
conditions  under  which  rectilinear  descent  is  possible.  Since 
6  is  constant  we  have  from  the  first  equation  of  motion, 

af(s)u  =  W  cos  0, 
which  gives 

=  W  cos6 

: 


The  second  equation  then  gives  as  a  value  of  the  traction, 

_    mW  cos  0  d  I    i    \     TJ7         .  /  b\ 

T  =  -  —  (-—  -)-PFcos0(tan0  —  ). 

2         ds\af(s)/  \  a/ 

It  is  possible  to  put  the  quantity  af(s)  =  Ky(z)A  in  a  form 
more  convenient  for  the  present  case.     We  shall  write 

Kv(z)=KvP(z), 

where  p(z)  denotes  the  ratio  of  the  density  of  the  air  at  altitude 
z  to  the  density  at  the  surface  of  the  earth,  and  Ky  is  the  value 
of  the  lift  coefficient  at  the  ground.  WTe  then  have 


d_(  __  L  _  \=A(—2  _  \jz__sin0  d/  i  \ 
ds\AKyp(z)>     dz\AKyP(z))  ds~     AKydz\p(z))' 


ds\af(s) 

The  traction  can  then  be  written 

\m  sin  B  d  (   i   \  ,  b~\ 

T=-Wcos6\     — -—  -y-^    +  tan0 —  . 
L  2AKV  dz\p(z)/  a\ 

Suppose  that  the  angle  of  descent  is  given  by  the  relation 

tan  8  =  -. 
a 

Before  making  the  simplification  that  results  from  this  assump- 
tion we  note  that  W/KyA  is  the  square  of  the  horizontal 
velocity  at  the  ground  for  the  given  angle  of  attack.  Denote 
it  by  F2.  Then 

V2  m  sin  20  d  I   i 


4  dz 

If  the  air  were  of  constant  density,  we  see  that  it  wuuld 
be  possible  for  a  machine  to  glide  in  a  straight  line  without 
any  tractive  force  being  supplied.  For  a  given  angle  of  attack 


34  THE  DYNAMICS   OF  THE  AIRPLANE 

there  is  a  single  angle  of  glide  for  which  this  is  possible,  namely, 
the  angle  given  above.  It  is  seen  to  be  the  angle  </>  in  the 
polar  diagram.  In  this  case  the  component  of  the  weight 
along  the  path  exactly  balances  the  drag,  and,  if  the  machine 
is  started  with  the  proper  velocity  along  this  course,  it  will 
maintain  the  course  without  change  of  velocity. 

With  the  air  varying  in  density  the  situation  is  quite  other- 
wise. We  have  now  for  the  square  of  the  velocity 

W  cos  S     W  cos  9 

Sll   

a/(s)       AKvp(z)' 

which  decreases  with  the  altitude.  If  the  angle  of  descent  is 
that  chosen  above,  the  component  of  the  weight  along  the 
path  balances  the  drag,  and  hence  an  outside  force  must  be 
used  to  cause  the  retardation.  The  necessary  retarding  force 
is  of  course  small,  on  account  of  the  slow  change  of  the  density 
of  the  air. 

34.  We  have  found  the  value  of  the  traction  with  the  assump- 
tion of  rectilinear  descent.  We  must  reverse  the  question  and 
assure  ourselves  that  by  starting  from  proper  initial  conditions 
rectilinear  descent  will  result  if  the  proper  traction  is  con- 
stantly furnished. 

We  start  with  the  equations 

mu  —  =  W  cos  0—af(s)u, 
as 

m    dli  rr>     ,     TT7-       «  7  ff     \ 

—  =  r+PFsm  d  —  bf(s)u, 
2  ds 

and  suppose  that  the  initial  conditions,  those  for  5  =  0,  satisfy 

the  relations 

W  cos6-af(s)u  =  o, 

IF  sin  0-bf(s)u  =  o, 
that  is,  initially, 

b  W  cose 

tan  6  =  -,    u  =  — 77-T— • 
a  af(s) 

Put 

_Wcos6 

af(S)    ' 


DESCENT   AND  ASCENT  35 

and  suppose  that  the  traction  is  given  constantly  by  the  expres- 
sion 

fj,  =  md^ 
2  ds' 

The  equations  of  motion  can  then  be  written, 


dd     T,7 

mu  —  =  W  cos  0 
ds 


["      u\ 
i  —   , 
v] 

m  d  f        \     TIT  b  ii\ 

—  —  (u-v)  =  W  cos  6  tan  6  — 
2  ds  a  flj 


a  v 

Making  use  of  the  definition  of  z>,  and  putting 

w  =  u  —  v, 
the  equations  take  the  form 

dd  ,,  s 

mu  —  =  —  awf(s)  , 
(ts 


2  as 
The  quantity  u  still  occurs  in  the  first  equation,  but  will  cause 

no  inconvenience.     The  initial  conditions  are  0  =  tan~1-,  w  =  o. 

a 

while  u  initially  has  a  definite  positive  value. 

Consider  an  interval  o  =  s  =  s'  so  small  that  within  it  w 
and  6  have  no  extrema,  and  therefore  their  derivatives  are 
not  zero  except  for  s  =  o.  Therefore  throughout  the  interval 
•w  and  6  are  mono  tonic. 

Suppose  w  were  increasing,  so  that  — ->o.    Then  since  wf(s) 

Q/S 

is  increasing,  and  both  terms  on  the  right-hand  side  of  the  second 
equation  are  initially  zero,  we  see  from  that  equation  that  d 

must  increase.     Hence  —  >o.     But  this  evidently  contradicts 

ds 

the  first  equation. 


36  THE   DYNAMICS   OF   THE   AIRPLANE 

Suppose  next  that  in  the  interval  considered  w  is  decreasing, 
and  therefore  negative,  since  it  is  initially  zero.  The  second 
term  on  the  right  of  the  second  equation  will  be  >o  in  the 

interval,  so  that  we  must  have  tan  0<-,  for  -7-  must  be  <o. 

a          ds 

Therefore  6  must  be  decreasing.     But  the  right-hand  side  of 

the  first  equation  is  now  positive.     Hence  -T->O.     There  is 

ds 

therefore  again  a  contradiction. 

It  follows  that  throughout  the  interval  we  must  have  w  =  o. 
Therefore 


= 


af(s) 

It  also  follows  that 

6  =  constant  =  tan"1  -. 

We  see  that  the  initial  conditions  that  we  have  assumed 
are  maintained  throughout  the  interval  o  =  s  =  s',  and  therefore 
by  a  process  of  continuation  will  be  maintained  throughout 
the  motion.  Therefore  rectilinear  motion  will  assuredly  result 
from  the  initial  conditions  and  the  value  of  the  traction  that 
we  have  assumed. 

35.  We  shall  next  consider  the  path  of  descent  that  is 
followed  when  the  machine  is  descending  with  the  propeller 
not  running.  The  equations  of  motion  are 

de 

mu  —  =  W  cos  8—af(s)u. 

m  —  =  2W  sin  6—2bf(s)u. 

We  have  already  demonstrated  that,  if  the  density  of  the  air 
remained  constant,  it  would  be  possible  to  glide  down  a  straight 
line,  provided  the  machine  started  in  the  correct  direction  with 
the  proper  velocity.  In  the  present  problem  we  shall  assume 
these  same  initial  conditions.  That  is,  for  s  =  o,  we  assume  the 
right-hand  sides  of  the  two  equations  to  have  the  value  zero. 
We  recall  further  that/(Y)  is  an  increasing  function. 


DESCENT  AND  ASCENT  37 

We  consider  as  before  an  interval  o  =  s  =  s'  within  which 
u  and  6  are  mono  tonic.  Suppose  u  increases;  then  T~>O, 

QS 

and  the  second  equation  shows  that  6  must  increase,  that  is, 

—  >o.  But  from  the  first  equation,  since  6  increases  cos  6 
ds 

decreases,  and  we  see  that  the  right-hand  side,  being  zero  initially, 
becomes  negative.  Hence  -r<d-  Therefore  there  is  a  con- 

CLS 

tradiction.  Hence  u  cannot  increase.  The  same  reasoning 
shows  that  u  cannot  remain  constant  throughout  the  interval. 

Suppose  next  that  u  decreases.  Hence  -j-<o-  The  quan- 
tity }(s)u  may  be  either  increasing  or  decreasing.  As  far  as  the 
second  equation  then  is  concerned,  we  see  that  9  may  be  increasing 

or  decreasing.     Suppose  it  is  increasing.     Then  ^->o.    In  order 

ds 

that  9  may  be  increasing  the  second  equation  shows  we  must 
assume  f(s)u  to  be  increasing.  The  first  equation  then  has  its 
right-hand  side  negative,  since  cos  0  decreases,  and  f(s)u 
increases.  This  gives  a  contradiction,  since  with  9  increasing 

we  have   -— >o.     Suppose,  now,  that  9  decreases;    then  f(s)u 
ds 

may  be  either  increasing  or  decreasing  so  far  as  the  second 
equation  is  concerned.  In  this  instance  -T-<O-  As  cos  0 

increases  the  first  equation  shows  that  f(s)u  must  be  increasing. 
But  no  contradiction  is  reached.  Therefore  u  and  0  both  decreas- 
ing are  the  only  possibilities.  Consequently  as  the  machine 
descends  both  u  and  0  decrease. 

It  is  apparent  that  the  right-hand  sides  of  the  equations 
having  once  become  negative,  u  and  0  must  continue  to  decrease. 
As  far  as  our  analysis  has  shown,  the  right-hand  sides  might 
subsequently  have  the  values  zero.  But  in  that  instance  the 
preceding  development  shows  that  they  must  then  again 
become  negative. 

We  have  thus  proved  that,  if  the  machine  is  started  down- 


38  THE  DYNAMICS   OF  THE  AIRPLANE 

wards  under  conditions  that  would  cause  a  rectilinear  glide 
if  the  air  density  remained  constant,  the  angle  of  descent 
becomes  continually  less,  and  the  velocity  likewise  diminishes. 
In  the  vicinity  of  the  earth  the  density  of  the  air  does  not  change 
very  rapidly.  Therefore  the  departure  from  the  straight  line 
of  the  path  that  a  machine  follows  when  gliding  is  not  great, 
if  the  descent  is  from  a  low  altitude.  It  is,  in  fact,  customary 
to  speak  of  rectilinear  glide  as  an  actual  state  of  possible  motion. 
36.  We  shall  assume  that  the  descent  is  from  a  low  altitude, 
so  that  the  course  can  be  regarded  as  rectilinear.  There  is  now 
equilibrium  between  the  air  reaction  and  the  weight,  so  that  we 
have  the  simplified  equations  of  motion, 


Wsmd  =  KxAV2+ksV2, 

where  we  are  using  the  values  of  the  various  coefficients  at  the 
surface  of  the  earth. 

The  angle  0  will  be  the  angle  <£  in  the  polar  diagram,  as  is 
seen  by  taking  the  quotient  of  the  last  two  equations.  It  will 
be  the  smallest  when  the  angle  of  attack  is  the  optimum  angle^ 
which  is  generally  between  7°  and  8°.  It  is  therefore  the  angle 
of  attack  to  be  used  in  case  it  is  desired  to  make  the  longest 
possible  glide  from  a  given  height. 

The  velocity  along  the  path  is  given  by 


If  we  let  V  be  the  horizontal  velocity  for  the  same  angle  of 
attack,  we  have 

V=vVcos  0, 

which  shows  that  the  velocity  along  the  curve  is  less  than  in 
horizontal  flight. 

Squaring  and  adding  the  equations  we  find 


DESCENT  AND  ASCENT  39 

Let  r  represent  the  distance  from  the  origin  to  the  point  on  the 
polar  corresponding  to  the  angle  of  attack.     Then  we  can  write 


A    r 

It  is  of  course  necessary  to  use  the  true  polar  for  this  purpose. 
While  the  latter  is  poorly  adapted  to  show  the  fineness  of  the 
machine,  because  it  is  sensibly  parallel  to  the  Ky-a,xis,  it  gives 
the  value  of  r  without  any  difficulty. 

37.  The  rapidity  of  vertical  descent  is  V  sin  6.  We  shall 
denote  it  by  v,  and  shall  find  the  angle  of  attack  for  which 
it  is  a  minimum.  For  such  an  angle  of  attack  the  machine 
would  require  the  longest  time  to  reach  the  earth  from  a  given 
altitude.  An  approximation  to  the  angle  in  question  can  be 
easily  obtained.  We  have 


v  =  V  sm  0=  7cos  0-sin  0=F-tan  0-cos%  0. 

For  the  values  of  0  that  occur  we  can  replace  cos  0  by  unity 
without  much  error,  and  therefore  have  approximately 

y=F-tan  0=F-tan  <£, 

where  0  is  again  the  angle  used  in  the  polar  diagram.     But 

T 

tan  0  =  —  ,  where  T  is  the  traction  necessary  for  horizontal 
W 

flight.     Hence,  approximately  a 


. 

~  w   w' 

where  P  is  the  power  necessary  for  horizontal  flight.  It 
follows  that  the  angle  of  attack  that  will  make  v  a  minimum 
is  close  to  the  angle  of  minimum  power,  that  is,  the  economical 
angle,  which  in  §  28  was  denoted  by  0:2. 

38.  Denote  by  as  the  angle  of  attack  that  renders  v  a 
minimum.  While  it  differs  little  from  #2,  it  is  of  interest  to 
determine  which  of  the  two  is  actually  the  larger,  and  to 
ascertain  5  such  a  relation  is  invariable  fbr  all  machines. 


40  THE  DYNAMICS   OF   THE  AIRPLANE 

We  have 

W  T 

v2  =  V2sm2  0  =  ^--sin20. 
A   r 

In  place  of  the  angle  <f>  let  us  use  the  ordinary  polar  angle, 
namely,  the  angle  that  the  radius  vector  makes  with  the  -K^-axis, 
and  which  we  denote  by  ft.  Since  sin  0  =  cos  ft,  we  see  that 
v  will  be  a  minimum  when  r/cos2  ft  is  a  maximum.  If  we  knew 
the  equation  of  the  polar  curve,  the  point  for  minimum  v 
would  be  obtained  by  equating  the  derivative  of  f/cos2  ft  to 
zero. 

We  next  consider  the  angle  that  gives  minimum  power. 
In  §  28  we  have  shown  that  we  can  write 


where  A  is  a  constant,  and  x  and  y  are  the  Cartesian  coordinates 
of  a  point  on  the  polar,  with  Q  for  origin.  When  transformed 
to  polar  coordinates  this  becomes 


—  \/       *      3  £        * 

To  find  the  angle  of  attack  for  minimum  power,  it  is  then 
sufficient  to  find  the  point  on  the  polar  for  which 

r  sin3  ft 
cos2  ft  ' 

is  a  maximum.     The  derivative  of  this  with  regard  to  ft  can  be 
written 

in2/? 


Return  to  the  point  on  the  polar  that  corresponds  to  the 
angle  as  for  minimum  vertical  velocity.  For  that  point  we 
have 

*(     r    \=Q 


DESCENT   AND   ASCENT 


41 


Furthermore,  this  value  of  (3  gave  a  maximum  to  the  function 
r/cos2  0.     Consequently, 

d        r 


according  as  0  is  greater  or  less  than  the  value  fa  corresponding 
to  0:3.  Now  we  know  that  az  is  not  much  different  from  «2, 
which  is  known  to  be  greater  than  ai,  the  optimum  angle. 
Assuming  then  that  a3>«i  we  see  that  on  the  part  of  the  curve 


FIG.  19. 

corresponding  to  a>a\  the  polar  angle  ft  is  a  decreasing  function 
of  a.  Suppose  then  thata>o;3.  It  follows  that  @<(3z,  and  con- 
sequently, 

dfr\' 

— I 1  >o. 

d/3\cos2  j(3/ 

If,  however,  ai<a<as,  then  0>03,  and  consequently 

d(     r     \ 

-   <o. 
djftVcos2  /?/ 

Since  all  the  other  quantities  that  appear  in  the  expressions 
that  we  have  obtained  for  the  derivative  of  r  sin3  /8/cos2  /3  are 
positive,  we  see  that  that  derivative  will  vanish  only  if  a<a3. 
Consequently  it  follows  that 


42 


THE   DYNAMICS   OF  THE  AIRPLANE 


To  make  the  proof  complete  suppose  that  Ms,  the  point 
corresponding  to  as,  is  to  the  left  of  MI,  corresponding  to  «i. 
Prolong  OMs  until  it  intersects  the  polar  in  the  second  point 
Ms,  for  which  the  corresponding  angle  of  attack  is  as'.  Then 
for  a,  any  place  between  as  and  as,  we  would  have  (3>Ps  and 
consequently 


FIG.  20. 


and  M2  could  lie  some  place  between  MS  and  M3'.  It  could 
not  lie  to  the  left  of  Ms  or  the  right  of  MS,  as  in  either  case 
j8</33,  and  accordingly 


This  would  give  us  the  opposite  of  the  result  obtained  by 
assuming  as  sufficiently  near  a2  to  assure  that  MS  is  to  the  right 
of  MI.  But  let  us  reverse  our  last  reasoning  and  assume  a2  as 
given.  We  see  at  once  that  MS  must  be  to  the  left  of  M2, 
the  point  used  in  §  28.  It  would  follow  that  as  is  not  almost 
equal  in  value  to  a2.  Therefore  it  must  be  that  a2<as  is  a 
relation  invariable  for  all  machines. 

2.  ASCENT 

39.  Suppose  that  the  machine  is  in  rectilinear  horizontal 
flight,  and  that  a  greater  admission  of  gasoline  is  given  to  the 


DESCENT  AND  ASCENT  43 

motor.  The  increased  traction  of  the  propeller  exceeds  the  air 
resistance;  the  velocity  of  the  machine  increases,  and  the 
lift  accordingly  exceeds  the  weight  of  the  machine.  The 
machine  starts  to  rise.  The  angle  of  attack  changes  as  a  result 
of  the  vertical  component  of  the  force.  The  upward  motion 
causes  a  greatly  increased  air  reaction  downwards  on  the  tail 
plane,  and  the  moment  of  this  force  turns  the  machine  in  such 
a  way  as  to  keep  its  axis  in  the  vicinity  of  the  flight  path. 
The  complete  discussion  of  the  action  of  the  machine  must 
be  based  upon  the  considerations  given  later  in  the  chapter 
on  stability.  Oscillations  take  place,  that  tend  to  die  out, 
and  there  results  what  can  be  called  a  state  of  steady  ascent, 
with  the  forces  on  the  machine  in  equilibrium,  and  the  axis 
tangent  to  the  trajectory.  As  the  altitude  of  the  machine 
increases  the  power  necessary  for  horizontal  flight  increases. 
Therefore  the  excess  of  power  of  the  motor  becomes  smaller, 
and  ultimately  the  machines  flies  horizontally. 

A  similar  behavior  results  if  the  admission  of  gasoline  is 
not  altered,  but  the  angle  of  attack  is  changed,  by  manipula- 
tions of  the  elevator,  in  such  a  way  as  to  lessen  the  power 
necessary  for  horizontal  flight.  The  excess  power  liberated 
will  be  absorbed  by  the  rising  of  the  machine. 

40.  The    Equations    of    Motion.— Let    GT    represent    the 
tangent    to    the    path    at    any 
instant,  G  being    the   center  of 
gravity   of    the    machine.      The 
forces  perpendicular  to   the  tan- 
gent   are    then    the    lift    com-      D 
ponent  of   the  air  reaction  and  a     ^ 
component  of  the  weight.     The 
forces  along  the  tangent  are  the 
drag    and    a    component  of  the 
weight,    both    in    a    downwards 
direction,   and    the    traction    of  FlG 

the    propeller    in    the    opposite 

direction.  Expressing  the  fact  that  the  forces  are  in  equilib- 
rium, we  have  the  equations 


44  THE  DYNAMICS   OF  THE  AIRPLANE 


T  =  Wsm6+KxAV2+ksV2. 

41.  The  Velocity  along  the  Path.—  If  the  angle  of  attack 
is  given,  and  the  direction  of  the  flight  path,  that  is,  the  angle  6, 
the  first  of  the  equations  of  motion  gives  us  at  once  the  in- 
stantaneous velocity.  It  is  convenient  to  express  this  in  terms 
of  Vy  the  velocity  for  horizontal  flight  at  the  same  angle  of 
attack.  Evidently 


It  is  apparent  that  as  6  approaches  90°  the  value  of  V 
approaches  zero  as  a  limit.  The  special  case  6  =  go0  requires 
extra  consideration.  In  order  that  the  machine  should  actually 
mount  vertically,  like  a  helicopter,  that  is,  6  =  90°,  and  V  not 
be  zero,  it  is  customary  to  state  that  the  angle  of  attack  must 
be  that  which  makes  Kv  vanish,  that  is,  the  angle  of  zero  sus- 
tentation.  Otherwise  there  would  be  a  force  perpendicular  to  the 
flight  path.  For  this  angle  of  attack  the  velocity  V  may  have 
any  value,  provided  we  furnish  sufficient  tractive  power.  But, 
as  noted  above,  for  a  given  angle  of  attack,  other  than  that  of 
zero  sustentation,  V  approaches  zero,  when  6  approaches  90°, 
and  we  can  think  of  the  machine  as  standing  vertically  in  the 
air,  with  any  angle  of  attack,  provided  the  traction  exactly 
balances  the  weight. 

42.  The  Force  of  Traction. — The  traction  at  any  instant 
is  given  by 

T  =  Wsm6+KxAV2+ksV2. 

The  traction  for  horizontal  flight  at  the  same  angle  of  attack  is 
T  =  (KxA+ks)V2. 


Hence, 


T-Wsme    V2 

—  =  — =  cos  6, 

T  V2 


from  which  we  derive 


DESCENT   AND   ASCENT 


45 


w 


A  simple  geometrical  construction  follows  at  once.  Con- 
struct a  rectangle  of  sides  T  and  W,  the  latter  being  vertical. 
Then  T  is  the  sum  of  the  projections  of  T  and  W  on  a  line 
making  an  angle  6  with  the  horizontal, 
that  is,  an  angle  equal  to  the  inclination 
of  the  flight  path  at  that  instant.  It  there- 
fore follows  that  T  is  the  projection  of 
the  diagonal  of  the  rectangle  on  this 
line.  The  diagonal  makes  with  the  vertical 
an  angle  0,  where  tan  0  =  $,  the  fineness  of 
the  machine.  Since  the  length  of  the  diag- 
onal is  r/sin.0,  and  since  it  makes  with 
the  line  representing  the  direction  of  the  flight  an  angle  90°  —  0  —  0, 
we  can  write 

T=j  sin  (0+0)  =  w  sin  (0+0) 
sin  0  cos  <t> 

The  results  that  we  have  derived  also  show  that  the  traction 
necessary  to  mount  is  the  same  that  it  would  be  if  the  machine 
were  subject  to  two  forces,  its  weight  and  a  horizontal  force  T. 

The  traction  will  be  a  maximum  when  the  flight  path  is 
along  the  diagonal  of  the  rectangle.  In  this  case  0  =  90°  — 0, 
and 


T 

FIG.  22. 


T  = 


W 


sin  0     cos  0 

If  we  take  .i5  =  tan0  as  a  medium  value  of  the  fineness, 
we  find  for  the  maximum  traction  i.oiXW,  that  is,  the  maxi- 
mum traction  to  be  experienced  in  ascending  is  i  per  cent 
greater  than  the  weight  of  the  machine.  As  long  as  the  angle 
of  ascent  is  less  than  90°  — 20  the  traction  is  less  than  the 
weight  of  the  machine.  This  angle  can  be  as  high  as  76°, 
when  the  fineness  is  close  to  .12.  As  the  angle  6  continues 
to  increase  from  this  point  the  traction  exceeds  the  weight, 
reaching  the  maximum  given  above  at  a  value  as  high  as  83° 
for  the  angle  of  ascent.  It  then  decreases  as  the  angle  in- 
creases to  90°,  on  account  of  the  rapidly  decreasing  speed, 


46 


THE   DYNAMICS   OF   THE   AIRPLANE 


when  it  again  equals  the  weight,  and  we  have  the  case  of  motion^ 
less  sustentation. 

It  is  obvious  that  for  ordinary  angles  of  ascent  the  traction 
will  not  exceed  the  weight. 

43.  The  Power  Necessary. — The  power  necessary  to  ascend 

On  the  diagonal  of  the  rectangle  used  in  the  last  section 

as  diameter  construct  a  circle, 
omitting  that  part  of  the  circle 
that  lies  below  the  horizontal  and 
the  part  to  the  left  of  the  vertical. 
On  the  vertical  lay  off  a  distance 
V  and  to  the  left  of  the  verti- 
cal construct  the  velocity  curve 
FVcos  0,  as  shown  in  Fig.  23. 
For  a  given  angle  of  ascent  the 
power  is  then  given  by  the  area 
of  the  rectangle  OABC. 

Let  the  diameter  of  the  circle  be  D,  and  7  the  angle  shown. 
Then 

P  =  D  cos  7-FVcos  0. 
Hence, 

_D-V    cos  7 
dO        2      Vcos  0 


=  -D-V  sin  yVcos  0  -?- 


sin  6. 


Evidently  dj/dd=—i.  so  that  the  condition  for  a  maximum 
(the  minimum  of  P  is  o,  for  B  =  90°,  for  then  V  =  o)  of  P  is 

tan  7  =  J  tan  0. 

For  this  relation  to  be  satisfied  it  is  evident  that  0<9O°  —  </>. 
Therefore,  the  angle  for  ascending  at  which  the  power  is  a 
maximum  is  less  than  the  angle  for  which  the  traction  is  greatest. 
If  61  is  an  angle  for  which  the  power  necessary  to  ascend 
is  greater  than  the  power  necessary  for  horizontal  flight, 
there  is  a  second  angle  02  for  which  the  required  power  is  the 
same  as  that  for  B\.  The  velocity  for  02  being  less  than  for  0i, 
the  traction  must  be  greater.  (We  note,  however,  that  we 


DESCENT   AND  ASCENT  47 

cannot  say  that  simply  because  02>0i  the  traction  corre- 
sponding to  62  must  be  greater  than  that  corresponding  to  61, 
because  the  traction  has  been  shown  to  have  a  maximum.) 

44.  The  Vertical  Velocity.  —  The  rapidity  with  which  the 
machine  is  momentarily  moving  vertically  is 

z;=F-sin  6. 
If  the  equation  of  traction  is  multiplied  by  V  we  can  write 

W-v  =  P-(KxA+ks)V*, 
or  in  terms  of  V, 

W-v  =  P-  (KA  +ks)  F3  cos3'*  6, 
which  can  again  be  simplified  into 


where  P  is  the  power  necessary  for  horizontal  flight. 
If  the  angle  of  ascent  is  small,  we  can  write 

P  —  P  P  —  P 

^  =  375     TTr     miles  per  hour  =  5  50  —  —  -  feet  per  second, 
Vv  W 

where  P  and  P  are  measured  in  horse-power. 

Now  P  is  the  useful  power  that  the  motor  is  developing, 
so  that,  providing  the  angle  of  mounting  is  small,  the  velocity 
of  ascent  is  equal  to  the  excess  of  power  over  that  required  for 
horizontal  flight,  divided  by  the  weight.  Further,  we  see 
that  for  a  constant  available  power,  the  velocity  of  ascent  is 
greatest  when  the  angle  of  attack  is  at,  that  is,  the  angle  for 
minimum  power  for  horizontal  flight.  Remembering  the 
results  that  were  obtained  for  minimum  vertical  velocity  in 
gliding,  we  see  that  an  angle  of  attack  that  gives  a  small  vertical 
velocity  in  gliding,  will  give  a  large  one  in  ascending,  provided 
we  have  a  constant  available  power. 

The  ideas  that  have  just  been  adduced  must  be  modified 
when  we  come  to  consider  the  propelling  plant.  Changing  the 
angle  of  attack  changes  the  velocity  and  traction.  This  alters 
the  number  of  revolutions  per  minute  that  the  propeller  must 
make  in  order  to  furnish  the  traction  at  that  velocity.  This  in 


48  THE   DYNAMICS   OF   THE   AIRPLANE 

turn  alters  not  only  the  efficiency  of  the  propeller,  but  also 
the  power  which  the  motor  is  developing.  Thus  in  practice 
we  do  not  have  a  constant  available  power.  Consequently 
the  best  angle  for  mounting  cannot  be  expected  to  be  exactly 
the  economical  angle. 

Another  approximation  is  useful.     We  have 

v  v 

?   fc^*V 

We  are  assuming  that  practically  we  can  take  cos0  =  i,  and 
can  therefore  take  8  =  sin  6.  In  degree  measure  we  can  there- 
fore write 

P-~P  P-~P 


approximately,  provided  the  result  is  a  small  angle.  In  this 
formula  the  excess  useful  power  is  expressed  in  horse-power, 
the  weight  of  the  machine  in  pounds,  and  the  velocity  in  hori- 
zontal flight  in  miles  per  hour. 

45.  We  return  to  the  general  considerations  of  §  39.  Suppose 
the  pilot  upon  leaving  the  ground  gives  full  admission  of 
gasoline,  developing  more  power  than  is  necessary  to  fly  hori- 
zontally near  the  ground  at  that  angle  of  attack.  The  machine 
starts  to  rise.  The  values  for  the  velocity,  traction,  and  power 
which  have  been  obtained  in  our  discussion  are  those  that 
exist  at  any  moment.  They  are  known  if  the  direction  of  the 
tangent  to  the  path  is  known.  The  general  problem  that  arises 
is  of  great  difficulty.  The  power  of  the  motor  for  a  specified 
number  of  revolutions  per  minute  decreases  as  the  machine 
gains  altitude,  because  the  diminishing  atmospheric  pressure 
decreases  the  mass  of  gas  mixture  that  is  in  the  cylinder  at 
each  explosion.  Further,  the  changing  density  of  the  air 
modifies  the  propeller's  action,  and  the  changing  velocity 
alters  the  number  of  revolutions  per  minute.  If  the  angle  of 
attack  is  not  changed,  the  curvature  of  the  path  decreases 
with  the  altitude,  and  the  machine  ultimately  flies  horizontally. 
This  occurs  when  the  power  the  motor  is  developing  multiplied 
by  the  efficiency  of  the  propeller  is  exactly  the  power  required 


DESCENT  AND   ASCENT  49 

for  horizontal  flight  at  that  altitude.  The  height  to  which  the 
machine  has  risen  is  called  the  ceiling.  It  is  dependent  upon 
the  angle  of  attack.  But  the  word  can  be  used,  without  danger 
of  confusion,  to  denote  the  height  to  which  the  machine  can 
rise  for  a  given  angle  of  attack,  or  the  maximum  of  these  heights, 
that  is,  the  greatest  height  at  which  the  machine  can  possibly  fly. 

As  the  machine  is  fitted  with  various  instruments  for 
measuring  height,  velocity,  time,  etc.,  a  record  of  the  flight 
can  be  made,  and  from  such  data  experimental  laws  can  be 
adduced.  Though  some  of  these  laws  may  appear  only  approx- 
imate, they  will  nevertheless  allow  us  to  obtain,  by  equations 
derivable  from  them,  other  results  of  sufficient  accuracy  to  be 
both  of  interest  and  value.  And  it  is  only  in  this  way  that  cer- 
tain problems  can  be  attacked. 

46.  Experimental  Law  of  Vertical  Velocity. — Experience 
shows  that  the  vertical  velocity  decreases  sensibly  as  a  linear 
function  of  the  altitude.  Suppose  the  machine  is  flying  at  the 
angle  of  attack  that  allows  it  to  reach  its  ceiling.  Let  h  be 
the  height  of  the  ceiling,  and  VQ  the  vertical  velocity  at  the 
ground.  Then 


where  z  is  the  altitude.  It  is  convenient  to  measure  the 
velocities  in  feet  per  second  and  //  and  z  in  feet. 

Experience  also  shows  that  the  initial  velocity  ^o  can  be 
calculated  very  approximately  from  a  knowledge  of  the  power 
of  the  motor,  the  weight  of  the  machine,  and  the  height  of  the 
ceiling.  Let  HQ  be  the  power  of  the  motor  at  the  earth,  using 
full  admission  of  gas.  Then  the  quantity 

W 

W==JT 
#o 

is  the  weight  per  unit  horse-power,  a  quantity,  like  the  loading, 
of  great  importance.  The  quantity  VQ  is  then  given  approx- 
imately by  the  relation 

h 

' 


50  THE  DYNAMICS  OF  THE  AIRPLANE 

For  example,  a  machine  weighing  7.59  pounds  per  horse- 
power, and  having  a  ceiling  at  24,500  feet,  will  'have  an  initial 
vertical  velocity  of  34  ft.  /sec. 

We  can  then  write 

h-z 
v  =  --  . 

95*- 

47.  Time  of  Ascent.—  We  have 
dz 

5 

Integrating  this  we  find 

*(«)-*.  log 


.- 

•   B) 


In  terms  of  common  logarithms  this  becomes 


h  .          i 

=  2. 303 --log 
VQ 


the  time  t(z)  being  expressed  in  seconds. 

48.  Determination  of  the  Ceiling. — The  result  which  has 
been  derived  for  the  time  of  ascent  can  be  used  in  a  very  simple 
way  to  determine  the  height  of  the  ceiling. 

We  have 


H)- 


Let  /(zi)  and  t(z%)  be  the  times  required  to  reach  altitudes 
0i  and  22,  respectively,  zi<S2,  and  put 


=  n. 
t(*i) 
Then 


Hence, 


DESCENT   AND  ASCENT  51 

Suppose  now  that  in  ascending,  the  pilot  notes  the  time 
t(zi)  when  he  reaches  any  chosen  altitude  z\,  and  that  he  then 
notes  his  altitude  22  when  the  interval  2t(z\)  since  leaving 
the  ground  has  elapsed.  We  have  then  n  =  2. 

Hence 


When  this  is  expanded  and  solved  for  h  we  find 


2Z-1  — 2? 

It  is  therefore  unnecessary  for  the  pilot  to  determine  the 
ceiling  by  flying  until  his  altimeter  indicates  that  he  is  no 
longer  gaining  altitude. 

Suppose  that  a  machine  requires  8  minutes  40  seconds  to 
reach  an  elevation  of  9850  feet,  and  that  in  double  the  interval, 
that  is,  in  17  minutes  20  seconds  it  has  reached  an  altitude  of 
15,500  feet.  We  then  have  for  the  height  of  the  ceiling, 

^9850? 23;loofeet. 

2X9850-15,500 


CHAPTER  IV 
CIRCULAR  FLIGHT 

i.  HORIZONTAL  TURNS 

49.  When  we  consider  the  question  of  controllability  of  an 
airplane  we  are  at  once  confronted  by  the  problem  of  turning. 
A  machine  must  be  equipped  with  controls  which  will  allow 
its  course  to  be  altered  at  the  will  of  the  pilot,  with  ease,  cer- 
tainty, and  safety.  It  is  found  at  once  that  the  successful 
solution  of  the  problem  gives  to  the  airplane  properties  that 
distinguish  it  from  other  vehicles  in  the  matter  of  turning. 

The  airplane  will  not  change  its  direction  from  a  given 
vertical  plane  unless  one  of  the  three  forces  acting  on  it  has 
a  component  perpendicular  to  that  plane.  Obviously  the  force 
that  can  give  this  component  is  the  air  pressure.  Thus  the 
solution  of  the  problem  will  be  obtained  by  making  the  air 
resistance  have  a  component  in  the  direction  towards  which 
it  is  desired  to  turn.  It  is  thus  apparent  that  the  plane  of 
symmetry  must  be  inclined  to  the  vertical,  that  is,  the  machine 
must  be  banked.  The  center  of  gravity  will  then  be  urged  in 
the  direction  towards  which  the  plane  of  symmetry  has  been 
tilted,  for  the  air  reaction  being  in  this  plane,  will  now  have  a 
component  in  that  direction.  If  at  the  same  time  the  proper 
turning  moment  can  be  given  the  machine,  it  can  be  made 
to  keep  its  axis  tangent  to  the  curve  described  by  the  center 
of  gravity,  and  this  evidently  is  necessary  for  a  proper  turn. 
Therefore  to  effect  a  turn  we  must  create:  (i)  a  centripetal 
force,  (2)  a  turning  moment. 

It  will  be  found  that  new  phenomena  arise  when  we  have 
deflected  the  machine  from  a  horizontal  straight  line  flight. 
The  machine  may  slip  sideways,  that  is,  in  a  direction  per- 

52 


CIRCULAR  FLIGHT 


53 


pendicular  to  the  plane  of  symmetry.  This  is  undesirable. 
A  turn  is  regarded  as  correct  when  the  instantaneous  direction 
of  motion  remains  always  in  the  plane  of  symmetry.  There  is 
then  no  slipping.  This  can  be  accomplished,  and  a  machine 
can  be  gotten  into  such  a  position  that,  with  controls  fixed, 
it  will  describe  a  horizontal  circular  path. 

50.  Equations  of  Motion. — Suppose  an  airplane  with  center 
of  gravity  at  G  is  flying  in  a  circle  of  radius  r  with  center  in 
the  direction  GC.  Let  6  be  the  angle  of  bank.  The  lift,  being 


'W 


FIG.  24. 


in  the  plane  of  symmetry,  is  represented  by  GL,  and  from  it 
we  must  obtain  both  the  sustaining  force  and  the  turning  force. 
Let  V  be  the  velocity  with  which  the  machine  is  moving.  We 
know  that  in  order  to  make  it  describe  the  circular  path 
there  must  be  a  force  toward  the  center  equal  to  mV2/r,  where 
m  is  the  mass  of  the  machine. 

We  therefore  have  the  three  equations  of  motion: 


trtV2 


T  =  KxAV2+ksV2. 

If  these  equations  are  satisfied,  the  center  of  gravity  will 
momentarily  be  describing  the  circle  in  question.  But  in 
order  to  make  the  motion  similar  to  that  of  a  rigid  body  rotat- 


54  THE   DYNAMICS   OF   THE   AIRPLANE 

ing  about  an  axis  there  must  be  a  turning  moment  applied  to 
the  machine  in  order  to  keep  the  axis  of  the  machine  tangent 
to  the  path.  The  method  of  producing  this  moment  will  be 
considered  later.  Furthermore,  it  is  known  from  the  principles 
of  rigid  dynamics  that  when  a  rigid  body  rotates  around  an 
axis  the  centrifugal  reaction  also  gives  rise  to  a  couple  tending 
to  displace  the  axis,  unless  the  axis  is  a  principal  axis  of  inertia. 
This  couple  would  manifest  itself  by  an  alteration  of  the  angle 
6.  It  must  be  combated  by  the  controls.  The  gyroscopic 
action  of  the  propeller  and  engine  also  come  into  play.  They 
also  will  be  disregarded. 

51.  The  Velocity  and  Inclination. — From  the  first  equation 
of  motion  we  obtain 

V 


where  V  is  the  velocity  for  rectilinear  flight  at  the  same  angle 
of  attack.  It  therefore  follows  that  the  velocity  in  a  circle 
is  greater  than  that  in  a  straight  line.  In  turning,  an  airplane 
must  therefore  increase  its  speed,  contrary  to  the  practice 
of  other  vehicles. 

From  the  first  two  equations  we  have 

mV2     V2 
tan  0  =  — —=— , 
Wr     rg 


where  g  is  the  acceleration  of  gravity.     In  terms  of  V  we  find 

V2 

tan0  =  —       — . 
rg  cos  0 

Hence, 

V2 
sm0  =  —  • 

For  a  given  radius  of  turn  there  is  therefore  only  one  inclination 
that  will  produce  a  correct  turn. 

Using  this  value  for  sin  0  we  obtain  for  the  velocity  the 
expression 


CIRCULAR  FLIGHT   '  55 

We  therefore  have  both  the  inclination  and  the  velocity 
expressed  in  terms  of  V  and  the  radius  of  the  turn.  While 
we  can  to  a  certain  extent  regard  the  radius  as  an  arbitrary 
element  in  the  turn,  we  find  at  once  that  it  is  limited.  The 
value  for  sin  6  must  be  less  than  unity.  This  shows  that  we 
must  have 

V2    W      i          m 
r>  —  =  — 


g      g   AKV    AKy' 

This  is  consequently  an  inferior  limit  to  the  radius  of  the 
turn  that  a  machine  can  make.  It  depends  upon  the  angle 
of  attack.  It  is  smallest  for  the  angle  of  attack  that  gives  the 
maximum  value  to  Ky.  It  is  obvious  that  as  the  radius  of  the 
turn  approaches  its  inferior  limiting  value  the  angle  6  approaches 
90°  and  the  velocity  V  approaches  infinity. 

In  practice,  however,  a  machine  has  an  inferior  limit  to  the 
radius  of  a  turn  it  can  execute  that  is  much  greater  than  the 
theoretical  one  which  has  been  derived.  For  long  before  the 
theoretical  limit  has  been  reached,  the  power  of  the  engine  will 
have  become  insufficient  to  furnish  the  necessary  velocity. 

52.  The  Traction  and  Power.  —  The  traction_and  power  are 
easily  expressible  in  terms  of  their  values  T  and  P  for  horizontal 
flight  at  the  same  angle  of  attack. 

We  have 


f    ~V2    cos  0* 
Hence 


i  — 


In  the  same  way 

P     TV        i 


p     TV 
so  that 


56  THE   DYNAMICS   OF   THE   AIRPLANE 

Expanding  the  quantity  on  the  right   we  have 

o   v  4      2i   F 

-4-2+- 
4  r2g2     32 

Hence 


Suppose  the  radius  of  the  turn  is  large.  Then  at  the  same 
angle  of  attack  the  increased  power  necessary  varies  approxi- 
mately as  the  inverse  of  the  square  of  the  radius. 

53.  The  results  that  have  been  derived  have  merely  shown 
how  the  correct  turning  force  can  be  obtained  by  giving  the 
proper  inclination  to  the  plane  of  symmetry.  It  is  necessary 
to  see  how  the  machine  can  be  gotten  from  its  horizontal  straight 
line  path  into  the  required  banked  position,  and  how  the  tend- 
encies to  depart  from  this  position  can  be  overcome. 

The  controls  that  are  used  for  the  purpose  are  the  rudder 
and  the  ailerons.*  Suppose  the  rudder  is  turned  towards  the 
right.f  A  moment  is  produced  turning  the  machine  in  that 
direction,  and  at  the  same  time  a  small  force  tending  to  make 
the  center  of  gravity  move  slightly  to  the  left.  The  ultimate 
effect  of  the  air  pressure  on  the  rudder  and  the  keel  surface 
is  to  start  the  machine  turning  towards  the  right.  This  causes 
the  left  part  of  the  machine  to  move  faster  than  the  right. 
Consequently,  the  air  pressure  on  the  left  is  greater  than  on 
the  right,  and  the  machine  starts  to  bank  towards  the  right, 
which  is  the  correct  direction  to  produce  a  turn  to  the  right. 
This  banking  is  not  equal  in  general  to  that  required  for  a 
correct  turn.  The  pilot  increases  the  degree  of  bank  by  the 
ailerons.  The  ailerons  on  the  left  are  depressed  and  those 
on  the  right  are  raised,  thus  producing  a  greater  lift  on  the  left 
wing.  The  machine  having  been  banked,  the  lift  on  the  wing, 
which  before  the  turn  was  vertical  and  just  equal  to  the  weight, 
now  departs  from  the  vertical,  so  that  the  component  that  acts 
in  the  direction  opposite  to  gravity  is  no  longer  sufficient  to 

*  For  a  description  of  the  ailerons  see  §  98. 

t  See  §§  99,  100  for  a  fuller  explanation  of  the  rudder  and  its  action. 


CIRCULAR  FLIGHT  57 

sustain  the  machine,  and  the  machine  will  start  to  Jive,  unless 
its  velocity  is  increased.  For  this  purpose  the  pilot  may  either 
increase  the  admission  of  gas,  or  alter  his  angle  of  attack,  so 
as  to  obtain  an  excess  of  power.  If  it  should  happen  that  his 
machine  is  "  tangent/'  that  is,  an  excess  of  power  cannot  be 
secured  in  either  of  these  ways,  he  cannot  turn,  without  de- 
scending to  a  lower  altitude,  and  thus  liberating  some  excess  of 
power. 

As  the  outside  of  the  machine  moves  faster,  the  drag  there 
will  be  greater  than  on  the  side  towards  the  center  of  the  turn. 
This  has  a  tendency  to  make  the  machine  turn  towards  the 
left,  and  it  must  be  overcome  by  the  controls.  The  turning 
moment  on  the  machine  necessary  to  keep  the  axis  along 
the  path  is  secured  from  the  rudder. 

We  have  seen  that  in  order  to  turn  correctly  it  is  necessary 
to  produce  a  turning  moment  and  a  centripetal  force.  In 
general,  we  can  say  that  the  function  of  the  rudder  is  to  produce 
the  necessary  turning  of  the  axis  of  the  machine,  and  the 
function  of  the  ailerons  to  give  the  banking  that  is  required 
to  produce  the  necessary  centripetal  force. 

It  is  also  by  means  of  the  ailerons  that  a  tendency  to  slip 
is  overcome.  If  the  machine  tends  to  slip  towards  the  outside 
of  the  curve  the  ailerons  on  that  side  are  depressed  and  those 
on  the  other  side  raised.  This  banks  the  machine  further 
towards  the  inside,  throwing  the  resultant  air  force  further  from 
the  vertical,  and  thus  increases  the  force  towards  the  center. 
In  case  the  machine  starts  to  slip  towards  the  inside  of  the 
curve  the  ailerons  are  manipulated  in  the  contrary  way. 

When  it  is  desired  to  straighten  a  machine  out  after  a  change 
of  direction  has  been  accomplished  it  is  necessary  to  make  opera- 
tions the  reverse  of  those  described  for  making  the  turn. 

When  making  a  turn  the  pilot  must  guard  against  heading 
upward,  because  the  loss  of  speed  may  be  too  great  to  insure 
stability,  unless  a  large  excess  of  power  is  available. 

It  is  evident  that  in  a  banked  position  the  function  of  the 
elevator  and  the  rudder  are  not  distinct,  but  have  closely 
related  effects,  depending  upon  the  degree  of  inclination. 


58  THE  DYNAMICS  OF  THE  AIRPLANE 

In  some  early  machines,  turning  was  effected  without 
banking  the  machine.  The  necessary  centripetal  force  on  the 
center  of  gravity  was  secured  by  the  air  pressure  on  vertical 
partitions.  When  the  rudder  was  in  a  neutral  position,  there 
was  no  pressure  on  these  surfaces,  but  when  the  rudder  was 
turned,  a  force  arose  that  produced  the  turn.  A  good  deal 
of  slipping  arose  in  such  a 'turn,  and  therefore  the  turn  could 
not  be  considered  as  correct.  Furthermore,  the  drag  on 
the  vertical  surfaces  reduced  the  speed  of  the  machine. 

2.  CIRCULAR  DESCENT 

64.  We  shall  next  consider  the  question  of  helical  descent, 
without  the  motor  running.  The  traction  necessary  to  over- 
come the  drag  of  the  machine  will  be  furnished  by  the  weight 
of  the  machine,  and  the  necessary  centripetal  acceleration  will 
be  derived  from  the  air  reaction,  the  machine  being  again  in  a 
banked  position.  We  shall  assume  that  the  air  density  can  be 
considered  constant  in  the  distance  through  which  we  are  con- 
sidering the  motion. 

55.  Let  AGB  represent  the  path  of  the  machine,  G  being 
the  position  of  the  center  of  gravity  at  any  instant.  Let 
r  represent  the  radius  of  the  cylinder  on  which  we  are  supposing 
the  helical  path  is  traced.  Let  CGD  be  the  horizontal  circle 
through  G.  The  weight  of  the  machine  is  represented  by 
GW  =  W,  drawn  vertical.  The  centrifugal  force  is  repre- 
sented by  GH  =  H,  outward  along  the  radius.  The  resultant  of 
these  two  forces  is  represented  by  GQ  =  Q,  in  the  plane  through 
the  axis  of  the  cylinder  and  point  G,  and  making  an  angle  0 
with  the  vertical.  The  air  reaction  must  then  be  represented 
by  GR,  opposite  in  direction  and  equal  numerically  to  GH. 
The  velocity  of  the  machine  is  represented  by  GV=V,  the 
horizontal  component  by  GVo  =  VQ.  Let  further  \l/  be  the  angle 
between  GV  and  GV0. 

In  order  that  the  turn  may  be  properly  executed  it  is  neces- 
sary that  the  axis  of  the  machine  coincide  instantaneosuly 
with  the  tangent  to  the  path.  Therefore  the  plane  of  sym- 


CIRCULAR  FLIGHT 


59 


me  try  is  determined  by  GR  and  GV.  The  lift  component  of 
the  air  reaction  lies  in  this  plane,  being  perpendicular  to  GF, 
and  the  drag  is  along  GV.  The  resultant  Q  of  W  and  H  also 
lies  in  the  plane  of  symmetry.  We  can  then  consider  that  the 
force  Q,  the  lift  L,  and  the  drag  D  are  in  equilibrium.  In  the 


FIG.  25. 

plane  of  symmetry  draw  a  line  perpendicular  to  GQ.  Let  the 
angle  between  it  and  GV  be  ^ '.  Since  the  lift  L  is  perpen- 
dicular to  GV,  we  have,  upon  resolving  forces  perpendicular 
to  and  along  GV}  the  following  equations: 


60 


THE  DYNAMICS  OF  THE  AIRPLANE 


as   the  equations   of  sustentation  and   traction,   respectively. 
The  angle  6  is  evidently  given  by  the  relation 

tan,=^2=zo?  ,  '.;•;• 

Wr       gr 

where  m  is  the  mass  of  the  machine.* 

56.  The  angles  0,  \J/  and  ^'  are  evidently  not  independent, 
and  the  relation  between  them  can  be  easily  found  by  spherical 
trigonometry.  Let  GZ  represent  the  vertical  through  G,  GX 


904-0 


FIG.  26. 


FIG.  27. 


the  tangent  to  the  circular  section  CGD,  and  GY  the  radius 
continued  outward.  Draw  GZ'  in  the  plane  of  FZ,  making  an 
angle  0  with  GZ.  Likewise  draw  GX'  in  the  plane  of  XZ, 
making  an  angle  \f/  with  GX.  Then  evidently  the  plane  of  GZf 
and  GX'  represents  the  plane  of  symmetry  of  the  machine. 
About  G  describe  a  sphere  of  unit  radius.  We  thus  have  a 
right  spherical  triangle  which  we  can  represent  separately  from 
the  axes,  as  shown  in  the  Fig.  27.  We  have  then 

cos  C  =  cos  0-cos  (90°+^). 

If  we  recall  the  definition  of  ^',  we  see  that  it  is  the  angle 
between  GX'  and  a  line  in  the  plane  of  GZ'  and  GX'  and  per- 
pendicular to  GZ'.  It  then  follows  that 


Devillers,  "La  Dynamique  de  PAvion,"  p.  135. 


CIRCULAR  FLIGHT 


61 


The  preceding  relation  then  becomes 

sin  \l/'  =  cos  6  sin  ^, 
from  which  we  note  in  passing  that 


We  can  also  find  the  inclination  of  the  plane  of  symmetry. 
This  inclination  is  the  angle  between  the  plane  of  symmetry 
and  the  vertical,  that  is,  the  angle  /  in  the  right  spherical 
triangle  we  have  constructed.  Therefore, 


tan/  = 


tan  B 
cos  \l/' 


This  relation  can  be  put  in  another  form  which  will  later 
be  useful.     We  have 


cos  I 


cos  \f/ 


cos 


Therefore, 


Hence, 


Vcos2  ^-ftan2  B     Vsec2  6 -sin2  ty 
_       cos  \f/  cos  6 
Vi—  sin2  \l/  cos2  6 


cos/  = 

cos  ^' 
cos  6 


cos 


cos  \f/f 


cos 


cos  /' 


57.  Other  Form  of  Equations  of  Motion.— We  can  put  the 

equations  of  motion  in  another  form. 
The  path  described  is  a  skew  curve, 
whose  osculating  plane  contains  the 
tangent  and  the  radius  of  the  cylin- 
der. The  principal  normal  is  there- 
fore along  the  radius;  the  binormal 
is  perpendicular  to  the  tangent  and 
the  principal  normal. 

We  shall  resolve  the  forces  along 
the  tangent,  principal  normal  and 
binormal,  represented,  respectively,  by  GX,  GY,  and  GZ. 


FIG.  28. 


62  THE  DYNAMICS   OF  THE  AIRPLANE 

As  the  weight  of  the  machine  acts  vertically  downwards, 
its  components  are  respectively, 

W  sin  ^,        o,         —W  cos  ^. 

Consider  next  the  air  reaction.  It  acts  in  the  plane  of 
symmetry,  which  passes  through  GX  and  makes  an  angle  7 
with  the  vertical  plane  XGZ.  The  lift  L  lies  in  this  plane, 
and  is  perpendicular  to  GX.  Therefore  its  components  are 

o,        L  sin  /.        L  cos  /. 
The  drag  is  along  OX,  and  its  components  are 
-D,  o,  o. 

Finally,  we  have  to  consider  the  centrifugal  force,  mVo2/r, 
where  VQ  is  the  horizontal  projection  of  the  velocity.  This 
force  is  along  the  radius,  so  its  components  are 


-—  ,  o. 

For  steady  flight  the  sum  of  the  forces  along  each  axis 
must  be  zero.  Hence  we  have: 

D  =  W  sin  \j/, 

mV02 
Lsm/  =  —  —  , 

L  cos  /  =  W  cos  \l/. 

If  we  divide  the  second  equation  by  the  third,  we  can  take, 
as  the  equations  of  motion  :  * 

D  =  Wsmt, 
LcosI  =  W  cos  \l/, 

tan/--  5V-. 

rg-cos  ^ 

58.  We  next  show  that  the  equations  of  motion  last  derived 
are  identical  with  the  first.  In  the  latter  we  replace  Q  by 
TF/cos  6,  and  insert  L  and  D  on  the  right-hand  sides  of  the 
equations  of  sustentation  and  traction,  respectively.  The 

*  Cowley  and  Evans,  loc.  cit,  p.  222. 


CIRCULAR   FLIGHT  63 

former  equations  then  become,  when  written  in  the  order  of 
those  in  the  last  paragraph  : 


COS0 

W 


If  now  we  make  use  of  the  relations: 

sin  \l/'  =  cos  6  sin  \f/, 

cos  \f/'  _  cos  ^ 
cos  6   cos  Iy 

tan  6  =  tan  7  •  cos  ^, 

which  were  derived  in  §  56,  we  see  that  the  equations  of  motion 
last  written  become  identical  with  those  obtained  in  the  last 
paragraph. 

59.  From  the  equations  of  sustentation  and  traction  we  have 


so  that  \l/'  is  equal  to  the  angle  of  glide  for  rectilinear  descent, 
and  is  therefore  uniquely  determined  by  the  angle  of  attack. 

We  shall  again  regard  r  as  an  independent  variable,  specify- 
ing the  conditions  of  flight,  as  in  the  case  of  circular  flight,  and 
determine  the  other  quantities  that  specify  the  motion,  namely, 
\j/  and  B  in  terms  of  it  and  ^',  which  we  have  just  seen  is  deter- 
mined by  the  angle  of  attack. 

When  we  substitute 

F0=Fcos  f, 

in  the  third  equation  of  motion  we  obtain 

V2  cos  2t 

tan  6  =  ---  -. 


64  THE   DYNAMICS   OF   THE   AIRPLANE 

Making  use  of  the  equation  of  sustentation  this  becomes 


rgKyA 
But 


cos  19' 
so  that  we  have  finally 

sin  6  =  — :    -  cos  \j/f  cos2  \f/, 


m 

r^-r  COS 


,/r      sin2  *'l 

L1    o5F*J' 


upon  making  use  of  the  relation  between  \f/,  \J/'  and  S. 

When  we  replace  sin2  \l/f  by  i  —cos2  ^',  and  cos2  0  by  i  —sin2  9 
the  last  relation  reduces  to 

.  o  -     w  cos  ^'   .  9  .      .     .  .     w          q  .  , 
sin3  0  --  —  -—-  sin2  0-sm  ^H  —  —  —  cos3  ty  =o, 


a  cubic  equation  for  sin  0,  in  which  all  the  coefficients  are 
known  as  soon  as  we  know  the  radius  of  the  helix  and  the 
angle  of  attack.* 

Put 

m  cos  ^ 


The  equation  then  reduces  to 

y?  —  ax2  —  x+a  cos2  ^  =  0, 

where  x  is  the  unknown,  and  a  and  \j/r  are  known.     This  equa- 
tion can  be  written 

(x  —  a)  (x2  —  i)  =  a  sin2  ^', 
or 

/  _ 

1          " 


We  construct  now  the  curve, 

_sin2// 


*  The  cubic  equation  for  sin  0  is  given  by  Devillers,  loc.  cit.,  p.  138.     His 
discussion  of  the  equation  is  quite  different  from  that  given  here. 


CIRCULAR   FLIGHT 


65 


which  consists  of  the  three  branches  shown,  asymptotic  to  the 
lines  #+i=o,  x—  i  =o.     Draw  finally  the  line 


which  passes  through  (0,1)  and  has  a  slope  of  —  i/a. 


FIG.  29. 

The  roots  of  the  cubic  then  correspond  to  the  abscissae  of 
the  points  of  intersection  of  the  line  and  the  curve.  It  is 
obvious  that  all  three  of  the  roots  are  real.  Let  them  be 
denoted  by  xi,  xz,  and  #3.  It  is  seen  that 

-  00 


Since  #  =  sin  0,  the  only  root  which  is  applicable  is  X2.     We 
thus  have  sin  0,  and  consequently  0,  uniquely  determined. 


66  THE  DYNAMICS  OF  THE  AIRPLANE 

The  diagram  can  be  easily  modified  so  as  to  allow  the  value 
of  0  to  be  directly  obtained.  Draw  the  curve  x  =  sin  0,  taking 
for  0  the  negative  ^-axis,  as  shown.  By  continuing  the  ordinate 
through  x  =  X2  until  it  intersects  the  curve  #  =  sin  d,  the  value 
of  B  can  be  immediately  obtained. 

It  is  interesting  to  consider  the  limiting  values  for  B  when  r 
approaches  zero  and  infinity,  respectively •.  The  corresponding 
values  of  a  are  evidently  oo  and  o.  The  slope  of  the  line  we 
have  used  becomes  —  oo  for  a  =  o,  and  evidently  the  corre- 
sponding value  of  X2  is  zero.  Thus  for  r  =  oo  we  have  6  =  o, 
and  the  descent  becomes  rectilinear. 

Let  r  =  o,  so  that  a  =00.  The  root  #2  is  then  determined 
by 

^sin2^ 

I—X22 

This  gives 

sin2  6  =  i  -sin2  f  =  cos2  tf/, 
and  therefore 

0  =  9o°-t//. 

This  is  a  maximum  value  for  the  angle  0. 

After  we  have  obtained  the  values  of  \l/r  and  6,  we  can 
at  once  get  ^,  the  angle  of  descent,  from  the  relation 

sin  \f/' 

sm  ^  —  —    —. 
cose 

In  the  limiting  case  r  =  o  we  have  sin  ^  =  i,  so  that  ^  =  90°. 
The  machine  in  this  case  descends  vertically,  rotating  about 
its  axis  as  it  descends.  The  value  of  /  is  seen  to  be  equal 
to  90°. 

60.  The  Velocity. — We  have  from  the  first  two  equations 

of  §  55 

Q 

Hence, 


CIRCULAR  FLIGHT  67 

Let  V  be  the  velocity  for  a  rectilinear  descent  at  the  same  angle 
of  attack.     From  §  36  it  follows  that 

V'2 


COS0' 

and  therefore  V  can  be  determined  as  soon  as  0  has  been  found. 
We  see  that  the  velocity  in  a  helical  descent  is  greater  than  in 
a  rectilinear  glide.  In  the  limit  for  r  =  o,  we  have  cos  0  =  sin  \l/'. 
Hence  the  limit  of  V  is  given  by 

V'2 


sin*' 

where  B  is  the  fineness. 

The  distance  that  the  machine  will  advance  downwards  for 
each  revolution  about  the  axis  of  the  cylinder,  that  is,  the  pitch 

of  the  helix,  is 

h  =  2irr  -  tan  \f/. 

In  the  limit  as  r  approaches  zero,  this  becomes  indeterminate, 
for  \f/  then  approaches  90°.  In  order  to  investigate  the  limit 
we  shall  express  both  r  and  \f/  in  terms  of  6  and  \l/' '.  We  have 
when  this  is  done: 

sin  \l/'  sin  \l/' 

tan  \(/ 


Vcos2  0  —  sin2  \//f     Vcos2  fy' — sin2  0 

From  §  59  we  find, 

_m  cos  \j/'  cos2  \}/'  —  sin2  0 
KVA        sin  0- cos2  0   ' 
Hence, 


,  _  irm    sin  2  ^'  Vcos2  \j/'  —  sin2  0 
1^^  sin  0  •  cos2  0 

In  the  limit  when  r  =  o  we  have  sin  0  =  cos  \j/'.    Hence  in  the 
limit  h  =  o. 

If  we  write, 

h 

-  =  27r  •  tan  \f/, 


THE  DYNAMICS   OF   THE  AIRPLANE 


and  note  that  when  r  approaches  infinity  the  angle  ^  approaches 
^',  we  see  that  the  graph  of  h  as  a  function  of  r  approaches  the 
line 

h  =  r-2ir  tan  \/'  =  2irB  •  r 


asymptotically.*     But  we  know  that  t'<t,  so  that  the  graph 
of  h  as  a  function  of  r  would  be  as  shown  in  Fig.  30. 


FIG.  30. 

Suppose  the  machine  is  descending  at  an  angle  of  attack 
such  that  the  fineness  is  .15,  and  that  the  radius  of  the  helix 
is  300  feet.  Then  we  have  for  the  minimum  the  machine  can 
advance  for  each  revolution 


5  =  282  feet. 

*  Devillers,  loc.  cit.,  p.  148. 


CHAPTER  V 

THE  PROPELLER 

61.  The  power  of  the  motor  with  which  an  airplane  is 
equipped  is  transformed  into  thrust  by  means  of  a  propeller. 
We  can,  in  fact,  define,  in  a  general  way,  a  propeller  as  an 
instrument  which  will  furnish,  by  the  air  reaction  upon  it,  a 
thrust  along  an  axis,  when  it  is  rotated  about  that  axis.     Con- 
siderable variation  in  design  exists  in  propellers  that  are  used 
at  the  present  time,  and  the  entire  field  is  one  in  which  con- 
tinued investigation  is  necessary.     There  are,  however,  basic 
principles  which  are  followed  in  propeller  design,  and  general 
laws  that  determine  their  behavior.     In  the  short  treatment 
that  is  given  here  only  the  more  general  aspects  of  the  problem 
are  considered. 

62.  The  study  of  a  propeller's  action  is  one  of  great  difficulty. 
It  involves  both  theoretical  and  experimental   considerations. 
It  is  easy  to  see  that  a  complete  solution  of  the  problem  by 
means  of  mathematical  processes  is  hardly  to  be  hoped  for. 
With  the  speed  with  which  a  propeller  rotates  the  problem  be- 
comes much  more  complex  than  that  of  the  action  of  an  aerofoil. 
The  compressibility  of  the  air  will  play  an  important  part. 
The  air  in  front  of  the  propeller  is  drawn  in  towards  it,  so 
that  the  hypothesis  that  the  propeller  acts  upon  air  at  rest 
will  be  only  very  approximate.     Nevertheless  theoretical  con- 
siderations  are  very  important.     They  can  give  us   at  least 
tentative  laws  that  we  can  subject  to  tests,  and  they  help  us 
to   discover   experimental   laws,   by  suggesting   what   may  be 
the  relation  between  various  observed  phenomena. 

63.  The  laws  that  we  have  applied  to  the  consideration  of 
the  action  of  an  aerofoil  furnish  a  basis  for  the  design  of  a 

eo 


70 


THE  DYNAMICS   OF   THE   AIRPLANE 


propeller,  and  will  be  used  in  the  following  consideration  of  a 
propeller's  action. 

Let  Fig.  310  represent  one  blade  of  a  propeller.  Consider 
a  section  AB.  The  shape  of  the  section  is  shown  in  Fig.  316. 
It  is  seen  that  the  section  resembles  that  of  an  aerofoil.  As 
in  aerofoils,  variation  exists  in  the  amount  of  camber  of  the 


FIG.  3 1  a. 


FIG.  316. 


upper  and  lower  surfaces.     The  general  plan  of  the  blade  also 
varies.* 

64.  Consider  the  air  reaction  on  a  section  of  the  propeller 
at  distance  r  from  the  axis.  Suppose  the  propeller  is  making 
n  turns  a  second,  and  advancing  along  the  axis  at  the  rate  V 
feet  a  second.  At  a  given  instant  the  section  has  therefore 
two  velocities,  namely,  a  velocity  u  =  2-n-nr  in  a  direction  normal 
to  the  axis,  and  the  velocity  V  along  the  axis.  The  resultant 

V  of  these  two  velocities  is 
represented  by  AC.  We  con- 
sider that  the  section  is  in- 
stantaneously moving  in  the 
latter  direction.  The  angle 
of  attack  is  therefore  the 
angle  </>.  The  resultant  air 
reaction  R  will  be  in  a  direc- 
tion near  to  the  normal  to 
It  will  have  two  components, 


FIG.  32. 
the  chord  of    the    section. 


*  For  a  discussion  of  various  designs  in  propellers,  see  the  report  by  W.  F. 
Durand  in  the  Third  Annual  Report  of  the  National  Advisory  Committee  on 
Aeronautics,  1917,  Report  No.  14,  entitled  "Tests  on  48  Model  Forms  of  Air 
Propellers  with  Analysis  and  Discussion  of  Results  and  Presentation  of  the  same 
in  Graphic  Form." 


THE   PROPELLER  71 

one  in  a  direction  opposite  to  the  rotation,  and  the  other  along 
the  axis.  The  first  component  opposes  the  motor  couple,  and 
the  second  produces  a  thrust  along  the  axis.  A  similar  analysis 
holds  for  each  element  of  the  blade,  and  we'  see  that  the  resultant 
effect  of  the  whole  propeller  is  the  production  of  a  couple  that 
must  be  overcome  by  the  motor,  and  a  thrust  along  the  axis 
of  the  rotation. 

65.  Geometrical  Pitch.— In  the  figure  let  0  =  0+0.  The 
value  of  6  depends  only  on  the  inclination  of  the  section,  which 
we  shall  call  the  setting  of  the  section.  A  consideration  of 
the  figure  leads  us  to  regard  the  chord  of  the  section  as  a  part 
of  a  geometrical  helix  of  radius  r,  that  makes  an  angle  6  with 
a  section  normal  to  the  axis  of  the  helix.  From  this  viewpoint 
one  revolution  should  make  the  element  advance  a  distance 

H  =  2irr  -  tan  6. 

This  is  called  the  geometrical  pitch  of  the  element.  It  will 
vary  from  element  to  element,  unless  the  angle  6  for  an  element 
is  connected  with  the  radius  of  the  element  by  the  relation 

6  =  tan-1—, 

2irr 

where  H  is  some  constant. 

There  are  then  two  classes  of  propellers:  those  of  constant 
pitch,  and  those  of  variable  pitch. 

(i)  Propellers  of  Constant  Pitch. — By  choosing  the  angle  6 
to  satisfy  the  relation  given  above  the  pitch  of  the  propeller 
will  be  made  constant.  The  angles  at  which  various  elements 
of  the  blade  are  set  are  easily  shown  graphically.  We  con- 
struct the  distance  H/2ir  as  an  ordinate,  giving  the  point  P, 
and  on  the  axis  of  abscissae  lay  off  distances  equal  to  different 
fractions  of  the  propeller's  radius.  The  corresponding  points 
are  joined  to  P.  The  lines  obtained  evidently  give  the  inclina- 
tion of  the  chords  of  the  various  sections  to  a  plane  normal  to 
the  axis  of  the  propeller. 

Suppose  the  propeller  is  of  constant  pitch,   and    that  it 


72 


THE   DYNAMICS   OF   THE   AIRPLANE 


advances  due  to  the  thrust  it  creates  at  the  velocity  V.     We 
have  then  for  the  angle  0,  Fig.  32, 

V 

~1  -- 


V 

-1- 

2irnr 


so  that  the  angle  of  attack  is 

H 

0  =  0-/3  =  tan~1  --  — 
2irr 


tan 


,  2irr     2irnr  nH  —  V 

-1 ^tan-1  - 


H-V 


(27rr)2n+HV 


FIG.  33. 

This  depends  upon  r,  so  the  angle  of  attack  is  different  for 
different  elements  of  the  blade,  being  the  smallest  at  the  tip, 
and  increasing  towards  the  axis.  This  fact  can  be  considered  an 
objection  to  a  propeller  of  constant  pitch,  although  the  ultimate 
decision  must  depend  on  experiment. 

(2)  Propellers  of  Variable  Pitch. — The  angle  0  can  be  chosen 
so  that  the  angle  of  attack  of  all  elements  will  be  the  same, 
by  merely  determining  6  by  the  relation 


0+  tan 


-1 


2irnr 


where  </>  is  constant.     The  manner  of  graphically  obtaining  the 
inclination  of  the  various  sections  is  shown  in  Fig.  34.     The 


THE   PROPELLER 


73 


ordinate  is  constructed  of  length  V/2irn.  Lines  are  drawn  to 
various  points  on  the  axis  of  abscissae  as  in  the  former  case. 
An  angle  equal  to  the  chosen  angle  of  attack  is  laid  off  from 
each  of  these  lines,  and  the  new  lines  obtained  represent  the 
chords  of  the  various  sections. 

It  is  to  be  observed  that  if  the  angle  of  attack  of  all  elements 
of  the  blade  is  the  same  for  a  certain  value  of  V  and  n,  it  need 
not  be  the  same  for  all  elements  for  other  values  of  V  and  n. 

Let  the  propeller  be  designed  for  a  velocity  Fo,  and  number 
of  rotations  no,  and  choose  <£0  as  the  angle  of  attack  of  all 
elements;  then  the  setting  of  an  element  is  given  by 


V 
2-mi 


FIG.  34. 

as  a  function  of  r.     Now  suppose  this  propeller  advances  at 
rate  V  and  makes  n  turns  per  minute.     The  angle  0  is  now 


2irnr 


and  the  angle  of  attack  is 


tan 


It  is  seen  that  the  necessary  and  sufficient  condition  that  this 
does  not  depend  upon  r  is 

nV0  —  noV  =  o, 
that  is, 

V=Vo 
n      no 


74  THE  DYNAMICS   OF  THE  AIRPLANE 

The  angle  of  attack  will  thus  again  be  the  same  for  all 
elements  of  the  blade,  only  if  the  advance  per  revolution  is 
the  same  as  it  was  before.  This  quantity  is  of  fundamental 
importance  in  considerations  regarding  a  propeller. 

66.  Thrust  and  Power.  —  In  order  to  calculate  the  thrust  that 
a  propeller  will  produce,  the  power  that  is  necessary  to  rotate  it, 
and  the  efficiency  with  which  it  is  working,  experiments  are  made 
on  model  propellers,  over  a  wide  range  of  forward  velocities,  and 
speeds  of  rotation.  From  the  data  thus  obtained  the  action  of 
a  full  size,  geometrically  similar,  propeller  can  be  deduced. 
We  proceed  to  a  discussion  of  the  conditions  under  which  we 
can  compare  the  action  of  similar  propellers. 

Consider  geometrically  similar  propellers  of  diameters  D 
and  DI,  respectively.  Let  them  be  advancing  at  velocities 
V  and  Vij  and  rotating  n  and  n\  times  a  second,  respectively. 
In  case  the  angles  of  attack  of  corresponding  sections  of  the 
two  propellers  are  the  same,  we  can  compare  the  action  of  the 
two;  and  it  is  quite  apparent  that  we  cannot  expect  to  find 
any  simple  relation  between  the  action  of  the  two  propellers 
unless  this  is  the  case.  The  condition  for  the  equality  of  the 
angles  of  attack  of  corresponding  sections,  always  under  the 
hypothesis  that  the  propellers  act  on  air  at  rest,  is  easily 
obtained.  The  settings  of  the  corresponding  sections  are  equal, 
since  the  propellers  are  geometrically  similar.  It  is  then  neces- 
sary that  the  angle  /3  be  equal  in  the  two  sections.  The  con- 
dition for  this  is  obviously 

V  =  Vi 
nr    n\r\ 

where  r  and  r  \  are  the  radii  of  the  sections.     This  reduces  to 


nD 

since  r/r\—D/D\  for  similar  sections. 

It  is  assumed  that  the  propellers  are  acting  under  conditions 
that  satisfy  this  relation. 

In  the  two  propellers  take  sections  near  those  already 
chosen,  the  new  sections  also  being  at  distances  proportional 


THE  PROPELLER  75 

to  the  diameters.  Consider  the  portions  of  the  two  blades 
thus  obtained  as  small  aerofoils.  As  they  are  engaging  the  air 
at  the  same  angle  of  attack  the  air  reactions  will  be  assumed 
to  make  the  same  angles  with  their  chords,  and  thus  with  the 
axes  of  the  propellers.  Let  the  air  reactions  be  dF  and  dF\. 
Then 

dF      ds-V'2 


where  ds  and  dsi  are  the  areas  of  the  two  sections  of  blade, 
and  V  and  V\   their  total  velocities  (resultants  of  V  and  2Trnr} 
and  Vi  and  2-n-rini,  respectively). 
Now 


in  virtue  of  the  similarity  of  the  propellers,  and 

V1  _    zirnr  sec  /3       nr       nD 
Vi      2^n\r\  sec  |8    n\r\     n\D\ 
Hence 

dF 


Let  dT  and  dT\  be  the  elements  of  thrust  furnished  by  the 
two   sections.     Since   the   reactions   make   equal   angles   with 

the  axes,  we  have 

dT      n2D* 


By  dividing  the  two  propellers  up  into  corresponding  small 
sections  and  adding  the  thrusts  of  the  sections,  we  have 

T 


for  the  relation  between  the  total  thrusts. 
This  relation  leads  us  to  write 


where  a  is  some  function  of  V/nD,  applicable  to  geometrically 
similar  propellers.  Such  a  relation  must  be  confirmed  by 
experiment.  This  question  will  be  considered  presently. 


76  THE   DYNAMICS   OF  THE  AIRPLANE 

Let  dR  and  dRi  be  the  resistance  to  rotation  of  the    two 
sections.     Then 

dR 


Let  dP  and  dP\  be  the  work  done  on  the  two  sections  in  a 
second;  then 

dP  =  2-wnrdR,        dPi  =  i-Kn\r\dR\. 
Hence, 

dP 
dPi 

We  are  thus  led  to  put  for  the  power  necessary  to  turn  the 
propeller 


where  /3  is  a  function  of  V/nD  applicable  to  geometrically  similar 
propellers. 

67.  The  results  obtained  can  be  put  in  another  form.     We 
have 

/^27^2\ 

D2. 


Since  a  is  a  function  of  V/nD,  the  quantity  a(n2D2/V2)  is 
also  a  function  of  V/nD.  Hence  we  are  led  to  write, 

T  =  3V2D2     (pounds), 

where  3"  is  a  function  of  V/nD. 
In  the  same  way  we  obtain 

P  =  (P  V3D2     (foot-pound-seconds) , 

where  (P  is  a  function  of  V/nD. 

68.  To  investigate  the  accuracy  of  the  results  that  have 
been  obtained  it  is  necessary  to  test  several  propellers  that 
are  geometrically  similar.  Consider  one  of  them.  The  thrust 
T  is  measured  for  a  large  range  of  values  of  V  and  n.  The 
quantity  T/V2D2  is  calculated  and  is  plotted  against  the 
argument  V/nD.  In  this  way  a  curve  is  obtained  that  repre- 
sents the  value  of  SI  for  the  first  propeller.  The  other  pro- 
pellers are  then  tested,  and  it  is  seen  how  nearly  identical  the 
curves  they  give  are  with  the  first. 


THE   PROPELLER  77 

In  the  same  manner  the  formula  obtained  for  the  power  is 
subjected  to  verification.* 

69.  Efficiency. — The  efficiency  of  a  propeller  is  the  ratio 
between  the  useful  power  it  yields  and  the  power  that  it  absorbs 
from  the  motor.  Let  it  be  represented  by  E.  Then, 

E=T'V 


550  -P* 

the  thrust  T  being  measured  in  pounds,  and  the  velocity  V  in 
ft./sec.  and  P  in  horse-power.  When  the  expression  obtained 
for  T  and  P  in  terms  of  V  and  D  are  used  this  becomes 


E  being,  as  indicated,  a  function  of  V/nD,  applicable  to  similar 
propellers. 

70.  Effect  of  Altitude. — The  thrust  that  a  propeller  will 
produce  and  the  power  necessary  to  turn  it  vary  directly  as 
the  density  of  the  air,  and  will  thus  depend  upon  the  altitude. 
It  is  apparent,  however,  that  the  efficiency  is  independent  of 
the  altitude. 

The  diagrams  that  will  be  given  for  a  propeller's  action 
are  for  the  surface  of  the  earth.  The  thrust  and  power  for  a 
given  altitude  will  then  be  found  by  multiplying  the  thrust 
and  power  at  the  surface  of  the  earth  for  the  same  velocity 
of  advance  and  number  of  rotations  per  minute  by  the  ratio 
of  the  air  density  at  the  given  altitude  to  the  density  at  the 
surface  of  the  earth. 

71.  Graphs  of  the  quantities  3",  (P,  E  for  a  model  propeller  f 

*  Results  of  experiments  of  this  sort  on  four  geometrically  similar  propellers 
of  diameters  30,  36,  42,  48  inches,  respectively,  are  given  in  Report  No.  14,  part  I, 
Figs,  u,  12,  of  the  Third  Annual  Report  of  the  National  Advisory  Committee 
for  Aeronautics,  1917. 

f  This  is  propeller  No.  i,  in  the  Report  of  the  Advisory  Committee  on  Aero- 
nautics above  referred  to.  The  curves  are  not  given  in  the  form  in  which  they 
occur  in  the  report.  The  density  of  the  air  at  the  surface  of  the  earth  is  intro- 
duced. The  constant  100  that  occurs  in  the  report  is  omitted  from  the  thrust 
curve.  The  power  curve  is  obtained  from  that  for  torque. 


78 


THE   DYNAMICS   OF   THE   AIRPLANE 


are  given  in  Fig.  35.     The  quantities  5"  and  (P  are  called  the 
thrust  and  power  coefficients,  respectively. 

As  an  example  of  the  use  of  the  graphs,  consider  a  full-scale 
similar  propeller  with  a  diameter  of  8  feet.  What  will  be  its 
thrust  and  the  horse-power  required  to  turn  it  at  an  altitude 
of  2000  feet,  if  it  is  advancing  at  a  velocity  of  72  miles  an  hour, 
and  making  1200  turns  a  minute? 


.0009  .0018 


FIG.  35. 


V 


We  have  7  =  105  ft./sec.,  n  =  2o.     Hence  —  =  .66.     From 

nD 

the  diagram  we  find  ^  =  .00054,  (P  =  .oooj.  The  ratio  of  the 
density  of  the  air  at  the  given  altitude  to  the  density  at  the 
surface  of  the  earth  is  .9.  We  thus  have 


=  35o  pounds, 


.9X.ooo7Xio53X82 

—  —  =  85  horse-power. 

550 


THE  PROPELLER 


79 


The  efficiency  can  be  immediately  obtained  from  the  graph, 
arid  is  E  =  .75.  It  can  also  be  computed  directly  from  the  values 
for  T  and  P. 

72.  A  problem  of  fundamental  importance  is  that  of  the 
adaptation  of  a  propeller  to  a  given  airplane  and  motor. 
This  question  can  be  well  studied  by  constructing  a  series 
of  performance  curves  for  the  propeller. 


60 


80 


100  120  140 

Velocity,  Mi./  hr. 

FlG.  36. 


Consider  the  propeller  of  8  feet  diameter  used  in  the  last 
section,  and  let  it  be  acting  at  the  surface  of  the  earth.  By 
giving  to  n  successively  the  values  800,  1000,  1200,  1400 
revolutions  per  minute,  the  thrust  and  power  can  be  plotted 
against  forward  velocity. 

The  results  for  the  thrust  are  shown  in  Fig.  36.  It  is  seen 
that: 

i.  For  a  given  number  of  revolutions  per  minute  the  thrust 
decreases  with  increasing  velocity  of  translation. 


80 


THE    DYNAMICS   OF   THE   AIRPLANE 


2.  For  a  given  velocity  of  translation  the  thrust  increases  with 
an  increasing  number  of  revolutions  per  second. 

These  results  are  to  be  expected.  For  a  reference  to 
Fig.  32  shows  that  for  n  fixed,  increasing  V  diminishes  the 
angle  of  attack  of  the  various  elements  of  the  blade,  and  hence 
decreases  the  resultant  air  reaction,  and  consequently  the 
thrust,  since  the  total  reaction  is  approximately  normal  to 
the  chord.  On  the  other  hand,  for  a  fixed  V,  increasing  the 
value  of  n  will  have  the  opposite  effect. 


100  120  140 

Velocity,  Mi,/hr. 

FIG.  37. 


The  curves  representing  the  power  are  shown  in  Fig.  37. 
It  is  evident  that: 

1.  For  a  given  velocity  of  rotation  the  power  absorbed  by  the 
propeller  decreases  with  increasing  forward  velocity. 

2.  For  a  given  velocity  of  advance  the  power  necessary  increases 
with  increasing  rotational  velocity. 

The  efficiency  can  be  found  from  Fig.  35.  For  convenience 
of  reference  the  following  table  of  values  for  V/nD  for  the 
values  of  V  and  n  used  is  given: 


THE   PROPELLER 


81 


v^* 

s^n 

800 

1000 

1  200 

1400 

60 

-56 

•45 

•37 

•32 

70 

.66 

•52 

•  44 

•37 

80 

•75 

.60 

•50 

•43 

go 

-84 

•  67 

•56 

.48 

100 

•94 

•75 

.62 

•53 

no 

1.03 

.82 

.69 

•59 

1  20 

1-13 

.90 

•75 

.64 

73.  Motor  Diagram. — In  order  completely  to  solve  the 
problem  of  adaptation  of  the  propeller  to  the  machine,  it  is 
necessary  to  know  the  manner  in  which  the  power  of  the  motor 
varies  with  the  number  of  revolutions.  Up  to  a  certain  limit 


150 

cJ 

a 

75 

^-— 

-^ 

s. 

/ 

X 

\ 

\ 

x 

X 

\ 

X 

x^ 

\ 

\ 

X 

X 

\ 

s 

600       800       1000       1200       1400       1600  R.P.M. 

FIG.  38. 

the  power  of  the  motor  is  approximately  proportional  to  the 
number  of  explosions  per  minute,  that  is,  to  the  number  of 
revolutions.  After  that,  due  to  choking,  the  power  decreases 
rapidly  with  the  number  of  revolutions.  A  diagram  giving  a 
motor's  performance  is  as  shown  in  Fig.  38.  This  diagram  is 
constructed  for  full  admission.  When  the  motor  is  throttled 
a  similar  curve  is  obtained,  lying  below  that  given. 

74.  Consider  now  the  conditions  under  which  a  given 
propelling  plant  will  propel  a  given  airplane.  Suppose  we  wish 
the  machine  to  be  flown  horizontally  near  the  earth  at  a  velocity 
of  80  miles  an  hour.  In  §  22  we  described  the  method  of 


82  THE  DYNAMICS  OF  THE  AIRPLANE 

plotting  the  traction  necessary  against  forward  velocity.  Sup- 
pose the  traction  is  400  pounds.  From  the  thrust  diagram 
for  the  propeller,  we  find  the  number  of  revolutions  per  minute 
that  the  propeller  must  be  making  in  order  that  it  shall  produce 
a  thrust  of  400  pounds  when  advancing  at  80  miles  an  hour. 
Suppose  the  value  of  n  thus  found  is  noo.  We  next  turn  to 
the  power  diagram  and  find  what  power  must  be  furnished  to 
the  propeller.  Suppose  it  is  85  horse-power.  We  finally 
consult  the  motor  diagram  and  see  what  power  it  will  furnish 
with  full  admission  if  running  at  noo  revolutions  per  minute. 
If  this  is  in  excess  of  85  we  can,  by  throttling  the  engine  properly, 
supply  to  the  propeller  exactly  the  power  which  will  make  it 
turn  with  the  proper  number  of  revolutions  that  will  give 
it  the  forward  velocity  and  traction  that  are  necessary  to  sustain 
the  machine. 

75.  While  the  curves  which  have  already  been  given  allow 
us  to  answer  questions  relative  to  the  adaptability  of  a  pro- 
peller to  a  given  motor  and  airplane,  it  is  possible  to  construct 
a  diagram  which  will  give  a  more  comprehensive  view  of  the 
problem. 

On  the  power  curves  we  construct  curves  of  equal  efficiency. 
For  example,  let  us  construct  the  curve  representing  an  efficiency 
of  60  per  cent.  For  n  =  Soo}  say,  we  find  from  Fig.  35  the 
values  of  V.  On  the  power  curve  for  800  rev./min.  we  locate 
the  points  corresponding  to  the  values  of  V.  We  do  this  for 
n  =  iooo,  1200,  1400,  etc.,  and  through  the  points  found  draw 
curves.  This  gives  us  the  60  per  cent  efficiency,  curves.  Simi- 
larly, we  construct  the  curves  for  E  =  6$  per  cent,  70  per  cent, 
75  per  cent.  Assuming  that  we  do  not  desire  any  regime  of 
operation  where  the  efficiency  falls  below  60  per  cent,  we  have 
a  diagram  which  gives  clearly  the  combinations  of  V  and  n 
which  will  give  us  an  efficiency  of  the  desired  amount. 

76.  On  this  diagram  we  next  construct  a  curve  which  repre- 
sents  the  functioning  of  the  motor   for   full   admission.     To 
do  this,  consider  the  performance  diagram  for  the  motor.     For 
a  given  value  of  n  read  the  power  the  motor  will  furnish,  and 
locate  in  Fig.  39  on  the  curve  for  the  same  value  of  n  the  point 


THE  PROPELLER 


83 


where  the  propeller  absorbs  that  same  power.  This  gives  us 
a  curve  which  intersects  the  two  limiting  efficiency  curves 
already  drawn.  Under  normal  conditions  motor  and  propeller 


175 


150 


125 


75 


50 


25 


L 


70 


.90     100     110     120 
Velocity  Mi./hr. 

FlG.  39. 


130 


140     150 


must  be  made  to  operate  for  values  of  V  and  n  that  lie  within 
the  triangular  area  which  is  determined  in  this  way. 

77.  From  the  diagram  last  made  it  is  easy  to  construct  a 
diagram  that  gives  the  maximum  useful  power  available  from 
the  motor-propeller  group.  Multiply  the  value  of  the  power 


84 


THE   DYNAMICS   OF   THE   AIRPLANE 


along  the  full  admission  curve  by  the  value  of  the  efficiency, 
and  plot  the  product  against  the  value  of  V.  This  gives  the 
curve  PM  shown  in  Fig.  40.  For  a  given  value  of  V  this  curve 


100 


75 


P  50 


i 


ff. 


1 


A 


/2\ 


Z 


/\ 


A 


40 


70 


100 


130 


FIG.  40. 


gives  us  the  greatest  useful  power  we  can  obtain  from  the 
propeller  attached  to  the  given  motor.  The  efficiency  curves 
are  carried  over  from  the  last  diagram  in  an  obvious  way. 
Also  curves  representing  the  number  of  revolutions  per  minute 
are  easily  constructed. 


THE   PROPELLER  85 

78.  To  answer  the  question  of  the  adaptability  of  the  motor 
and  propeller  to  the  airplane,  we  plot  on  the  last  diagram  the 
useful  power  necessary  to  propel  the  airplane,  as  a  function 
of  V  (see  §24).  This  gives  us  the  curve  PA.  We  plot  only 
that  amount  of  it  within  which  operation  would  be  safe.  Call 
the  extremities  of  the  curve  a  and  b.  The  curve  will,  in  general, 
intersect  PM  in  two  points  a'  and  b' '.  Thus  the  motor-propeller 
with  full  admission  will  propel  the  machine  in  horizontal  flight 
at  two  different  velocities.  The  degree  of  adaptability  of  the 
motor  and  propeller  to  the  airplane  depends  upon  the  relative 
positions  of  a  and  a',  b  and  b'.  If  b  is  beyond  b',  as  shown, 
the  motor  and  propeller  are  unable  to  get  the  speed  out  of  the 
machine  that  its  construction  would  safely  allow.  If  b'  were 
beyond  b,  the  motor  and  propeller  could  develop  a  velocity  that 
would  be  dangerous. 

The  greatest  velocity  of  ascent  will  be  attained  when  the 
excess  power  available  is  a  maximum.  This  corresponds  to 
a  velocity  of  about  90  miles  an  hour  in  the  case  represented  by 
the  diagram  given. 


CHAPTER  VI 
PERFORMANCE 

i.  CEILING 

79.  In  §  48  we  have  given  a  discussion  of  the  height  to 
which  a  machine  can  fly.     That  height,  which  we  called  the 
ceiling,  depends  upon  the  angle  of  attack.     The  maximum  of 
these  heights  for  all  angles  of  attack  we  called  the  true  ceiling 
of  the  machine.     A  reference  to  §  44  would  lead  us  to  expect 
that  the  true  ceiling  would  be  reached  when  the  angle  of  attack 
is  the  angle  of  minimum  power,  for  then  it  would  seem  that 
there  would  be  the  greatest  surplus  power  available  for  climb- 
ing.    This  conclusion  depends,  however,  upon  the  supposition 
of  constant  available  power,  a  condition,  which  as  indicated 
in  §  44,  does  not  exist.     As  a  matter  of  fact,  the  true  ceiling  is 
attained   for    an   angle   of   attack   more   nearly   equal   to   the 
optimum  angle,    and  sometimes  even  smaller.*     In  order   to 
determine  the  height  of  the  ceiling  by  the  method  given  in  §  48, 
it  is  necessary  to  make  an  ascent  of  some  height.     We  shall 
now  consider  the  question  of  determining  the  height  of  the 
ceiling  directly  from  the  known  motive  power  of  the  machine, 
its  fineness,  etc. 

80.  If  the  ceiling  is  to  be  as  high  as  possible  for  a  given 
machine  and  motor,  it  is  necessary  that  the  propeller  be  well 
adapted  to  the  two.     For  suppose  that  the  machine  has  risen 
to  the  greatest  possible  height.     The  motor  will  be  running 
with  a  certain  definite  number  of  revolutions  per  minute.     If 
this  number  is  not  that  which  gives  the  greatest  possible  power 
from  the  motor,  and  the  value  of  V/nD  is  not  that  which  gives 

*  The  Sorbonne  lectures  of  Professor  Marchis,   Spring  semester,   1919.        i 


PERFORMANCE  87 

the  greatest  efficiency  to  the  propeller,  it  is  evident  that  there 
is  a  lack  of  proper  adaptation  in  the  various  elements  of  the 
machine. 

In  the  calculations  that  we  shall  make  of  the  theoretical 
ceiling  in  terms  of  the  motor  power,  propeller  efficiency,  and 
fineness  of  the  machine,  we  shall  assume  that  there  is  a  com- 
plete adaptation.  We  shall  therefore  obtain  a  quantity  which 
will  in  practice  exceed  the  performance  of  the  machine. 

Let  Po  be  the  power  of  the  motor  at  the  ground,  in  horse- 
power, and  \LZ  the  ratio  of  the  height  of  the  barometer  at  altitude 
z  to  its  height  at  the  surface  of  the  earth.*  Then  the  motor 
power  at  altitude  z,  in  foot-pound-seconds,  is 


Let  E  be  the  propeller  efficiency,  which,  of  course,  varies 
as  the  machine  rises.  But  in  accordance  with  what  was  stated 
above  we  shall  assume  that  it  has  attained  its  maximum  value 
at  the  time  the  machine  ceases  to  rise.  The  useful  power 
available  at  the  ceiling  is  therefore 


The  traction  necessary  for  horizontal  flight  is  BW,  where 
B  is  the  fineness,  and  W  the  weight.  Therefore  the  power 
necessary  is 

Pm  =  B'W-Vh) 

where  V*  is  the  horizontal  velocity,  in  feet  per  second,  at  the 
ceiling. 

Consequently,  at  the  ceiling  we  have 


The  fineness  B  is  independent  of  the  altitude,  but 


p* 

*  The  remainder  of  this  section  is  taken  directly  from  the  lectures  of  Professor 

Marchis. 


88  THE  DYNAMICS  OF   THE  AIRPLANE 

where  VQ  is  the  horizontal  velocity  at  the  surface  of  the  earth, 
and  pg  the  ratio  of  the  density  of  the  air  at  altitude  z  to  the 
density  at  the  ground. 

Inserting  the  value  of  Vh  and  dividing  by  Po-E,  we  have 

B'W-Vo     I 


as  the  relation  which  must  be  satisfied  at  the  ceiling. 
If  we  put 

,/r_B'W'V0 

~ 


we  have  a  number  characteristic  of  the  complete  machine. 
Into  it  enters  the  power  of  the  motor,  the  efficiency  of  the 
propeller,  the  total  weight  of  the  machine,  the  fineness,  as  well 
as  the  horizontal  velocity  at  the  ground.  It  would  again 
seem  from  this  expression  that  the  angle  of  attack  to  reach 
the  ceiling  should  be  that  which  renders  BVo  a  minimum, 
that  is,  the  economical  angle.  But  on  account  of  the  con- 
siderations adduced  in  §  79,  the  fineness  and  velocity  for  the 
optimum  angle  are  used. 

If  we  give  to  If  a  succession  of  values,  such  as  .8,  .6,  .4, 
.3,  .2,  which  will  cover  those  that  occur  with  customary  machines, 
and  for  each  such  value  of  M,  compute  the  value  of  the  quantity 

M 

M*    7?' 

for  a  succession  of  values  of  z,  differing,  say,  by  1000  feet, 
until  we  reach  a  value  of  z  for  which  the  quantity  in  question 
is  zero,  we  shall  have  the  values  of  the  ceiling  that  correspond 
to  the  various  values  of  M.  In  this  way  we  have  a  table  that 
will  give  the  theoretical  ceiling  for  a  machine,  as  soon  as  we 
know  its  characteristic  number. 

If,  for  example,  we  take  M  =  .4  we  find  for  2  =  17,000  feet, 

M 
»2  --  £  =  .526-.  4X1.308  =  .003. 

Therefore  we  can  take  17,000  feet  to  be  the  approximate  theo- 
retical ceiling. 


PERFORMANCE  89 

Another  remark  may  be  added.  The  weight  W  includes 
the  weight  of  fuel.  This  continually  decreases,  so  that  M 
decreases.  The  ceiling  thus  continually  increases  as  the  time 
increases,  until  such  part  of  the  fuel  remains  as  the  pilot  desires 
to  have  at  his  disposition  in  the  descent.  If  we  include  in  W 
the  weight  of  the  normal  fuel  carried,  the  calculation  we  have 
made  gives  the  approximate  height  to  which  the  machine  rises 
before  its  course  becomes  sensibly  horizontal. 

81.  Supercharge.*  —  If  it  is  desired  to  make  a  long  flight, 
a  quantity  of  fuel  in  excess  of  the  normal  is  added.  This 
increases  W  and  thus  M,  and  therefore  lowers  the  ceiling. 
A  similar  condition  exists  if  any  extra  load  is  carried,  though 
in  the  first  case  the  surplus  load  continually  disappears,  while 
in  the  second  instance  that  is  not  the  case,  unless  perhaps  the 
machine  is  such  a  machine  as  a  bomber. 

We  shall  investigate  the  change  in  the  height  of  the  ceiling 
produced  by  a  supercharge.  At  the  ceiling  we  have,  as  the 
equation  of  power, 


The  equation  of  sustentation  gives  us 

W  =  PhKvAVJ?. 

Let  us  assume  that  p/,  =  ju/»  as  an  approximation.!    Dividing 
the  two  equations  we  find 


B'KyA      ' 

If  we  assume  that  E  is  constant,  we  see  that  Vh  is  constant. 
Therefore  the  velocity  at  the  ceiling  is  independent  of  any 
supercharge  in  weight  that  the  machine  may  be  carrying. 
A  reference  to  the  equation  of  power  then  gives 

W 

—  =  constant. 


*  Devillers,  loc.  cit.,  pp.  173-177. 

f  For  the  accuracy  of  this  approximation  see  the  Appendix,  §  4. 


90  THE   DYNAMICS   OF   THE   AIRPLANE 

Suppose  now  that  W  represents  the  normal  weight  of  the 
machine  and  h  its  normal  ceiling.  Let  the  machine  be  super- 
charged to  a  total  weight  W,  and  let  Jj  be  the  ceiling  for  this 
weight.  We  have 


Therefore 

W 

As  all  the  quantities  on  the  right  are  known,  this  equation  will 
determine  /M>  and  consequently  hf. 

For  example,  suppose  a  machine  with  a  normal  weight  of 
1500  pounds  has  a  ceiling  of  20,000  feet.  Let  the  machine  be 
supercharged  to  weigh  2400  pounds.  What  will  be  the  new 
ceiling?  We  have  /**  =  .468.  Hence, 

^  =  .468  ^  =  .748. 
1500 

The  new  ceiling  is  then  approximately  at  8000  feet. 

2.  RADIUS  OF  ACTION  * 

82.  Another  problem  of  interest  is  that  of  calculating  the 
distance  to  which  a  machine  can  fly.     In  addition  to  the  other 
characteristics  that  must  be  taken  into  account  we  must  now 
consider  the  total  amount  of  fuel  carried  and  its  rate  of  con- 
sumption.    Assumptions  must  again  be  made  that  will  render 
the  results  only  approximately  correct,  but  nevertheless  they 
will  be  of  value  as  a  basis  of  estimates  as  to  performance. 
Let   Pt  =  power  of  the  motor  at  time  /  (/  =  o  at  the  start) ; 
Q  =  total  weight  of  fuel,  gasoline  and  oil,  at  time  of 

departure; 

Qt  =  weight  of  fuel  at  time  /; 

m  =  weight  of  fuel  consumed  per  horse-power  per  hour ; 
W  =  weight  of  machine  at  time  of  departure; 
B  =  fineness  of  the  machine; 
Wt  =  W  —  Qi  =  weight  of  machine  at  time  /. 
*  Devillers,  loc.  cit.,  Chapter  XII. 


PERFORMANCE  91 

Assuming  that  the  rate  of  consumption  of  fuel  per  horse- 
power is  independent  of  the  altitude,  we  have 


,~  ti. 

dQt  =  -~  —  at, 

3600 

as  the  consumption  in  time  dt,  the  second  being  taken  as  the 
unit. 

The  useful  power  is  given  at  any  instant  by  Wt-B-V.  In 
order  to  obtain  the  power  in  terms  of  the  performance  of  the 
motor  we  must  assume  a  constant  efficiency  for  the  propeller. 
We  shall  take  it  to  be  .75. 

The  useful  power  developed  by  the  motor  in  foot-pound- 
seconds  is  therefore  .75X550XP/.  We  consequently  have 

.75X550  XP««=T 

Hence, 


.75X550 
and  therefore, 

mWlBV 


3600  X. 75X55° 
=  mWtBV 
1,365,500 

Let  L  represent  the  horizontal  distance  traversed.  The 
machine  in  reality  will  be  Continually  rising,  due  to  decrease 
in  weight  from  fuel  consumption.  However,  except  at  the 
start,  it  will  be  practically  horizontal.  We  take  therefore 
dL  =  Vdt.  The  relation  between  the  element  of  path  and  fuel 
consumption  is  therefore 

,r     1,365,500  ^(^1,365,500      dQt 
mB      'Wt          mB    ~'W-Qt' 

Integrating,  and  noting  that  at  t]ie  start  Q;  =  o,  and  at  the 
instant  of  arrival  Qt  =  Q,  we  have 

,     1.365.500 ,          W 

T  —  _  '*"*     *J '  *J         locr    — 

J^t  —  ^rt^  s~\* 

mB  W  —  Q 

It  is  apparent  from  this  that  the  greatest  distance  can  be 
covered  with  a  given  amount  of  fuel  if  flight  takes  place  at 


92  THE  DYNAMICS  OF  THE  AIRPLANE 

the  angle  that  gives  the  smallest  value  to  B.  Therefore  the 
optimum  angle  should  be  used.  During  the  flight  the  machine 
consequently  flies  at  its  ceiling,  §79. 

We  shall  assume  #2  =  5.5  and  £  =  .12.  When  we  change  to 
common  logarithms  and  to  miles  we  find: 

L-gooo  logio pr,  approximately. 

*~w 

From  this  it  is  apparent  that  the  distance  that  a  machine 
can  fly,  under  the  suppositions  we  have  made  as  to  efficiency 
of  propeller,  fineness,  and  consumption  of  fuel,  depends  only 
on  the  ratio  of  the  weight  of  fuel  with  which  the  flight  is  started, 
to  the  total  weight  at  the  time  of  departure. 

In  order  to  cover  a  great  distance  with  a  machine  without 
replenishment  of  fuel  it  is,  of  course,  necessary  to  greatly  super- 
charge the  machine  at  the  start.  This  lowers  the  ceiling,  as 
was  stated  in  the  last  section,  and  it  is  necessary  to  know 
that  the  ceiling  has  not  been  lowered  to  such  a  point  that 
flight  would  be  dangerous. 

Let  us  assume  that  a  flight  of  3000  miles  is  desired.    Hence, 


from  which 


and  therefore 

Q-u 
W~ 

The  weight  of  fuel  carried  must  then  be  approximately  equal 
to  the  weight  of  the  remainder  of  the  machine. 

Suppose  the  values  of  m  and  B  for  a  machine  are  not  those 
used  to  obtain  the  formula  for  L.     We  should  then  multiply 

C  j*         12 

the  value  of  L  given  by  the  formula  by  —  X1^-. 

m      n 


PERFORMANCE  93 

83.  Let  us  now  consider  a  flight  that  consists  in  going  a 
certain  distance  and  returning  to  the  starting  point,  the  com- 
plete flight  to  consume  the  entire  fuel  supply.  It  is  evident 
that  more  fuel  will  be  consumed  in  the  going  part  of  the  trip 
than  in  the  return.  We  desire  to  know  the  amount  of  fuel 
that  can  be  consumed  in  the  first  part  of  the  journey,  and  leave 
assured  the  possibility  of  return. 

Let  WQ  be  the  weight  of  the  machine  without  any  fuel 
(dead  weight),  Q  the  total  weight  of  fuel  at  the  start,  Qg  the 
part  of  the  fuel  that  will  be  expended  in  going,  and  Qr  the  part 
expended  in  returning.  The  going  part  of  the  trip  can  be  thought 
of  as  that  of  a  machine  with  dead  weight  of  W+Qr  and  a  fuel 
load  of  Qy.  The  returning  trip  can  be  regarded  as  that  of 
a  machine  of  dead  weight  WQ  and  fuel  load  Qr.  As  the  distances 
traversed  are  the  same  we  have  from  the  formula  of  the  last 
section, 

Qr  Q* 


which  can  be  written 

Q-Q,_Q* 
W-Qg    W 

From  this  we  find 


Therefore, 


14/ 


since  the  only  admissible  root  is  one  less  than  W. 

If  we  develop  the  expression  for  Q0)  we  find  we  can  write 

Ql^-L.1    £  + 

Q     28  W 

which  gives  the  excess  over  half  the  total  fuel  which  can  be 
consumed  before  the  return  trip  is  started.* 

*  The  method  of  §  83  can  be  extended  to  determine  the  performance  of  a 
bombing  plane,  or  a  machine  carrying  freight.  In  such  cases  there  is  a  definite 
and  considerable  decrease  of  weight  at  a  certain  point  in  the  journey,  for  instance 
the  place  where  the  return  trip  starts.  See  Devillers,  loc.  cit.,  pp.  185,  186. 


CHAPTER  VII 
STABILITY  AND   CONTROLLABILITY 

84.  We  come  now  to  the  consideration  of  a  question  which 
is  of  the  greatest  importance  in  the  actual  practice  of  flying, 
and  of  the  greatest  interest  as  a  problem  in  dynamics.     This 
question   is   that   of   stability   and    controllability.     We   shall 
begin  the  discussion  by  setting  forth  some  general  ideas  in^ 
volved  in  the  problem,  so  as  to  make  clear  exactly  what  it  is 
that  is  desired. 

In  a  broad  way  the  term  stability  relates  to  those  properties 
which  an  airplane  must  possess  in  order  that  it  be  "  air- worthy/' 
that  is,  that  it  will  be  able  to  fly  without  too  great  an  element 
of  insecurity  and  danger.  When  we  recall  the  fundamental 
principle  by  which  flight  is  possible,  namely,  an  equilibrium 
between  tractive  force  of  a  propeller,  air  resistance  on  the 
machine,  and  gravity,  and  remember  that  the  air  resistance 
is  delicately  dependent  upon  speed  and  aspect  of  the  machine 
with  reference  to  the  wind,  while  gravity  is  a  force  entirely 
Beyond  our  control,  we  see  that  the  problem  will  be  one  into 
which  a  thorough  inquiry  must  be  made.  Once  the  equilibrium 
between  the  forces  is  destroyed,  will  it  be  possible  to  re-establish 
it?  The  air  is  never  absolutely  calm;  and  at  different  altitudes 
and  in  different  localities,  eddies  and  gusts  of  different  natures 
and  varying  magnitudes  will  be  encountered,  so  that  flight 
under  the  ideal  conditions  that  we  have  thus  far  assumed  will 
never  exist.  Therefore,  if  secure  flight  is  possible,  it  will  be 
accomplished  by  a  more  or  less  frequent  recurrence  of  states 
of  losing  and  regaining  equilibrium. 

85.  As  the  question  turns  primarily  on  the  air  reaction,  we 
can  seek  to  maintain  equilibrium  between  the  forces  in  two 

94 


STABILITY  AND   CONTROLLABILITY  95 

general  ways.  We  can  construct  the  machine  in  such  a  way 
that  it  will  inherently  possess  properties  that  will  make  it 
stable,  and  we  can  equip  it  with  movable  control  surfaces  that 
enable  the  pilot  to  alter  the  air  forces  and  thus  assist  in  re- 
establishing the  equilibrium.  These  same  control  surfaces  will 
also  be  the  means  by  which  the  pilot  maintains  command  of 
the  machine,  directing  it  along  the  course  that  he  desires  to 
follow.  Thus  we  have  the  two  qualities,  stability  and  con- 
trollability, both  necessary  conditions,  related  and  dependent 
upon  each  other,  and  as  we  shall  see,  to  a  certain  extent  incom- 
patible. 

86.  In  order  to  be  inherently  stable  a  machine  must  auto- 
matically maintain  the  same  attitude  towards  the  relative  wind, 
for  this  will  tend  to  keep  the  air  reaction  unaltered.     When 
gusts  are  met,  the  machine  will  thus  tend  to  head  into  them, 
and  if  this  tendency  is  very  strong  it  may  be  difficult  for  the 
pilot  to  keep  the  machine  flying  in  the  course  he  desires.     In  a 
general  way,  the  greater  the  inherent  stability  the  more  difficult 
it  will  be  to  make  the  machine  respond  to  the  controls.     It 
will  tend  to  combat  changes  in  its  course.     In  some  instances 
a  great  sensitiveness  to  controls  is  a  necessity.     Thus  in  the 
case  of  a  battle  plane,  where  rapid  maneuvering  is  essential, 
the  pilot  must  be  able  to  get  a  quick  and  pronounced  effect 
by  moving  his  controlling  surfaces.     The  degree  to  which  a 
machine  possesses  inherent  stability  accordingly  depends  upon 
such  things  as  size  and  purpose.     While  it  is  a  quality  that 
all  machines  must  possess  to  a  certain  extent,  still  a  condition 
can    exist    that    could    be    described    as    "  too    stable."     The 
machine  then  could  be  controlled  only  at  the  expense  of  con- 
siderable fatigue  to  the  pilot,   and  furthermore,  it  might  be 
an   uncomfortable   vehicle,    owing    to    rapid   oscillations    that 
would  arise  when  it  encountered  gusts. 

87.  In   order    to   deal   with    the   problem   of   stability   by 
mathematical  analysis,  greater  precision  must  be  given  to  our 
definition.     In  fact,   we  can  distinguish  between  stability  in 
the  general  sense  that  has  been  considered,  and  in  the  special 
restricted  mathematical  sense  to  which  we  proceed,  and  which 


96  THE   DYNAMICS   OF   THE   AIRPLANE 

will  be  developed  to  some  extent  in  the  next  chapter.  There 
can  easily  be  a  difference  of  opinion  as  to  the  degree  in  which 
it  is  necessary,  or  even  desirable,  that  a  machine  should  possess 
mathematical  stability.  On  the  other  hand,  it  is  agreed  that 
a  machine  which  is  mathematically  unstable  in  certain  respects 
would  be  a  very  unsafe  and  undesirable  one. 

88.  The  question  of  stability  is  one  that  can  enter  into 
most  dynamical  situations.     It  relates  to  the  effect  of  a  dis- 
placement from  a  position  of  equilibrium.     Such  a  displacement 
gives  rise  to  changes  in  the  forces  acting.     If  the  effect  is  to 
restore  the  state  of  motion  to  that  which  existed  prior  to  the 
disturbance,  the  equilibrium  is  said  to  be  stable;   if  the  system 
tends  to  depart  farther  and  farther  from  the  prior  state,  the 
equilibrium  is  said  to  be  unstable;   if  the  system  is  indifferent 
to  the  change,  and  maintains  a  state  of  motion  near  to  that 
which  would  have  existed  if  the  disturbance  had  not  taken 
place,  the  equilibrium  is  said  to  be  neutral.     As  an  example 
of  stable  and  unstable  equilibrium,  consider  the  motion  of  a 
marble  down  a  hollow  inclined  pipe.    Let  it  be  on  the  inside. 
It  will  follow  the  lowest  element,  and  if,  in  its  motion,  it  be 
slightly    deflected,    it    will,    after    oscillating  back  and    forth, 
regain  its  former  state  of  motion:    its  equilibrium  is  stable. 
Suppose,  on  the  other  hand,  that  the  marble  is  on  the  outside 
of  the  pipe,  and  rolling  down  the  top,  which  we  shall  consider 
slightly  flattened.     A  small  disturbance  will  cause  it  to  depart 
from   its   path    and   fall   from   the   pipe:    the   equilibrium   is 
unstable.     As  an  example  of  neutral  equilibrium,  let  the  marble 
roll  down  an  inclined  plane.     A  small  lateral  disturbance  will 
cause  it  to  follow  a  course  near  that  it  would  have  followed 
had  it  not  been  deflected :  its  equilibrium  is  neutral. 

89.  The  example  of  stability  which  has  been  given  illustrates 
two   distinct  ideas   that   enter   into  stability.     After   the  dis- 
placement has  taken  place,  there  must  exist  a  force  tending 
to  restore  the  former  condition.     If  such  a  force  exist,  there  is 
said  to  be  static  stability.      The  effect  of  this  force  will  in 
general  be  such  as  to  make  the  system  with  which    we  are 
dealing  return  to  the  position  of  equilibrium,  and  then  depart 


STABILITY  AND  CONTROLLABILITY  97 

from  it  in  the  opposite  sense.  In  this  way  oscillations  will  arise. 
If  these  oscillations  die  out  as  the  time  increases,  there  is  said 
to  be  dynamic  stability. 

We  shall  in  this  chapter  give  a  development  of  the  simpler 
aspects  of  the  problem  of  stability  of  an  airplane,  mainly  those 
connected  with  static  stability.  The  consideration  of  dynamic 
stability  is  reserved  for  the  next  chapter.  A  general  under- 
standing of  the  means  by  which  stability  and  controllability 
are  obtained  can  be  secured  by  a  simple  analysis.  It  is  only 
by  such  a  preliminary  procedure  that  we  can  construct  models 
and  make  the  proper  and  elaborate  experiments  that  will 
furnish  the  data  for  an  investigation  of  dynamical  stability. 

90.  In  our   first   analysis   we   shall   consider   the  effect   of 
rotation  upon  an  airplane.     To  distinguish  different  types  of 
rotation  we  draw  three    axes  in  the   machine  meeting  at  the 
center  of  gravity,  one  perpendicular  to  the  plane  of  symmetry, 
one  in  the  plane  of  symmetry,  parallel  to  the  propeller  axis, 
for  instance,   and   the   third  perpendicular   to   these   two.     A 
rotation  about  the  first  axis  is  described  as  pitching,  about 
the  second  as  rolling,  about  the  third  as  yawing.     We  shall 
consider   the   effect   on   stability   of   rotations   of   these   three 
types.     The  general  rotation  that  a  machine  can  experience  is 
a  combination  of  all  three.     The  more  elaborate  investigation 
in  the  next  chapter  covers  that  case.     It  is,  however,  by  con- 
sidering separately  the  three  types  of  rotation  that  one  is  led 
to  develop  principles  of  design  that  make  stability  possible. 

We  shall  first  consider  the  effect  of  pitching,  which  gives 
rise  in  its  simplest  form  to  what  is  called  the  problem  of  longi- 
tudinal stability. 

LONGITUDINAL  STABILITY 

91.  We  can  say  in  an  approximate  way  that   longitudinal 
stability  depends  primarily  on  the  manner  in  which  the  center 
of  pressure  moves  as  the  angle  of  attack  varies.     Consider  a 
plane  surface.     Suppose  it  were  the  sustaining  member  of  a 
machine.     In  horizontal  flight  the  resistance  R,  the  traction  J1, 


98 


THE  DYNAMICS   OF  THE   AIRPLANE 


FlG.  41. 


and  the  weight  W  meet  at  a  point,  say,  the  center  of  gravity. 
Suppose  some  influence  tended  to  rotate  the  machine  in  such 
a  way  as  to  increase  the  angle  of 
attack.  The  resistance  R  recedes  from 
the  attacking  edge,  remaining  sensibly 
parallel.  A  moment  then  arises  that 
tends  to  return  the  machine  to  its  origi- 
nal position.  It  is  therefore  statically 
stable. 

Suppose  now  that  we  had  for  sus- 
taining member  a  single  cambered  sur- 
face. With  increasing  angle  of  attack 
the  center  of  pressure  tends,  as  a  rule, 
to  approach  the  edge  of  attack,  and 
vice  versa.  Thus  the  air  pressure  tends 
to  create  a  couple  that  will  increase  the 
displacement,  and  we  would  expect  the  machine  to  be  static- 
ally unstable. 

92.  An  analysis  of  this  sort  should  be  pursued  further. 
Consider  the  surfaces  of  a  biplane.  For  different  angles  of  attack 
suppose  that  the  line  of  action  of  the  result- 
ant air  pressure  were  drawn  in  the  plane  of 
symmetry.  This  gives  a  one-parameter  family 
of  straight  lines.  They  envelop  a  curve,  called 
the  metacentric  curve.  For  the  angles  of 
attack  that  are  used,  this  curve  is  concave 
to  the  front,  as  shown  in  the  figure.  The 
point  of  tangency  of  the  metacentric  curve 
and  the  line  representing  the  air  reaction  is 
called  the  metacenter.  As  the  angle  of  attack 
increases,  the  metacenter  describes  the  meta- 
centric curve  in  the  sense  A  to  B. 

Suppose  the  machine  is  in  horizontal  flight. 
Let  the  propeller  pass  through  the  center  of  gravity  G;  then  the 
machine  being  in  equilibrium,  the  air  reaction  R  must  also  pass 
through  G.  Suppose  the  machine  pitches  so  as  to  increase  the 
angle  of  attack.  The  metacenter  moves  to  m',  the  line  of  action  of 


R' 


STABILITY  AND   CONTROLLABILITY  99 

the  air  reaction  to  R'.  A  couple  of  moment  R'xGP  is  created, 
where  P  is  the  foot  of  the  perpendicular  from  G  on  Rf,  and  R' 
represents  the  magnitude  of  R' .  This  will  be  a  restoring 
couple  if  G  is  below  K,  the  intersection  of  R  and  R'\  other- 
wise, it  will  tend  to  increase  the  displacement.  Therefore  there 
will  be  a  condition  of  equilibrium  if  G  is  below  the  metacentric 
curve.  The  metacentric  curve,  however,  in  general,  lies  too 
low  for  this  condition  to  exist.  Stability  must  therefore  be 
obtained  by  some  added  feature.  This  feature  is  a  tail  plane. 

93.  Tail  Plane. — Consider  a  plane  surface  to  the  rear  of 
the  sustaining  surface.  Let  ab  be  a  section  of  it.  Suppose 
that  the  air  reaction  on  it  is  originally  zero,  that  is,  that  it  is 
in  the  bed  of  the  wind.  Let  G  be  the  center  of  gravity. 
Suppose  the  machine  is  rotated  about  the  center  of  gravity 


FIG.  43. 

through  an  angle  0;  then  the  tail  plane  takes  a  position  a'b', 
inclined  at  an  angle  6  to  the  relative  wind.  The  air  reaction,  /•, 
on  it  will  be  practically  normal  to  it,  and  can  approximately 
be  represented  by  ksdV2,  where  k  is  the  constant  given  in  §  3, 
and  5  is  the  area  of  the  tail  plane.  There  thus  arises  a  restoring 
moment  of  amount 


where  d  is  the  distance  shown  in  the  figure.  The  air  reaction 
on  the  main  sustaining  surface  likewise  has  a  moment,  about  G. 
The  sum  of  the  two  must  be  a  restoring  moment,  in  order  that 
the  tail  plane  function  in  the  desired  manner.  Similar  con- 
siderations must  hold  for  a  motion  that  tends  to  decrease  the 
angle  of  attack. 

It  is  seen  at  once  that  a  problem  of  great  importance  is  the 
determination  of  the  proper  area  for  the  tail  plane,  and  its 


100  THE   DYNAMICS   OF   THE  AIRPLANE 

distance  from  the  sustaining  surfaces.  The  angle  at  which  it 
is  set  is  also  important.  It  is  found  that  it  must  make  a 
smaller  angle  with  the  relative  wind  than  the  main  plane.* 
The  tail  plane  does  not  act  as  an  independent  plane,  but  is 
greatly  influenced  by  the  wash  of  the  main  planes.  Its  actual 
behavior  must  be  determined  by  experiment  on  different 
combinations.  Investigations  of  this  sort  have  been  conducted 
by  Eiffel,  who  has  found  that  the  wash  from  the  front  surface 
has  the  same  effect  as  decreasing  the  angle  of  incidence  of  the 
tail  plane.  Thus,  if  the  tail  plane  is  apparently  in  the  bed 
of  the  wind,  it  is  in  effect  behaving  as  though  it  were  at  a  nega- 
tive incidence,  and  therefore  has  a  downward  pressure  exerted 
upon  it. 

94.  The  Elevator. — For  allowing  control  of  the  machine, 
and  increasing  longitudinal  stability,  it  is  fitted  with  an  elevator. 
This  is  the  plane  attached  to  the  rear  of  the  tail  plane,  and 


FIG.  44. 

movable  about  a  horizontal  axis.  By  means  of  it  the  angle 
of  attack  of  the  machine  is  altered.  Suppose  the  machine  is 
in  horizontal  flight  with  the  elevator  in  the  neutral  position 
ab.  Let  it  then  be  turned  through  the  angle  6  into  the  posi- 
tion be.  There  arises  a  force  /,  which  we  can  represent  with 
sufficient  accuracy  by 

,     ,  /T70        sin  6 

r'=ks'V2 


.4 +.6  sin  0' 

where  k  is  a  constant  and  s'  the  surface  of  the  elevator.     Suppose 
the  center  of  gravity  is  at  G'.    Let  Gb  =  d\   then  assuming  the 

*  Bothezat,  "fitude  de  la  Stability  de  1'aeroplane,"  p.  98. 


STABILITY  AND  CONTROLLABILITY 


101 


force  rr  as  normal  to  the  elevator,  we  have  for  the  moment 
created : 

, ,    k   ,T79J      sin  26 
M  =  -sV2d-  — . 

2  .4+. 6  sin  0 

The  variation  of  this  force  with  the  value  of  0  is  shown  by  a 
consideration  of  the  function 

sin  20 


•4-f-.6  sin  0 

This  function  reaches  a  maximum  between  35°  and  40°. 
It  therefore  follows  that  an  elevator  should  not  be  given  an 
inclination  to  exceed  something  like  30°. 

The  moment  caused  by  turning  the  elevator  into  the 
position  shown  in  the  figure  will  have  the  effect  of  rotating 
the  machine  in  such  a  direction  as  to  increase  the  angle  of 
attack.  In  the  meantime  the  moment  of  the  air  reaction  on 
the  sustaining  surfaces  changes.  The  machine  will  rotate  to  a 
position  such  that  the  moment  of  all  forces  (propeller  traction, 
air  reaction  on  sustaining  members,  tail  plane,  elevator)  passes 
through  the  center  of  gravity.  If  the  motor  power  is  at  the 
same  time  properly  altered,  the  machine  will  fly  at  a  different 
angle  of  attack. 

The  means  for  changing  the  position  of  the  elevator  are 
secured  by  equipping  it  with  a  lever  arm  perpendicular  to  its 
surface,  to  the  end  of  which 
a  wire  is  attached,  which  in 
turn  runs  to  the  cock-pit. 
To  turn  the  elevator  may 
require  considerable  muscu- 
lar effort  on  the  part  of 
the  pilot.  This  effort  can 
be  lessened  by  properly  bal- 
ancing the  elevator.  The 
figure  shows  schematically 
the  method  of  doing  this. 
Let  aABb  be  the  fixed  tail 
plane,  and  ABcde  the  elevator,  pivoted  about  AB.  It  is 


d 

FIG.  45. 


102  THE  DYNAMICS  OF  THE  AIRPLANE 

obvious  that,  by  this  disposition  of  the  axis,  the  air  reaction 
can  be  kept  fairly  near  the.  axis  so  that  the  moment  required 
to  turn  the  elevator  will  be  lessened.  The  resultant  air  reaction 
must  not,  however,  in  any  position  of  the  elevator  pass  through 
AB,  for  the  pilot  must  always  be  able  to  feel  a  tautness  in  the 
controls.* 

95.  It  is  seen  that  we  have  considered  only  the  question 
of  static  longitudinal  stability,   that  is,  the  question  of  the 
existence  of  a  couple  tending  to  return  the  machine  to  its 
original  position.     Granting  that  such  a  couple  exists  we  cannot 
ascertain  the  ultimate  effect  of  the  oscillations  it  will  produce 
without  knowing  its  magnitude  as  related  to  the  angle  through 
which  the  machine  has  been  turned.     We  see  also  that  while 
the  rotation  is  going  on,  the  instantaneous  angle  of  attack  of 
every  element  of  the  wing  is  changing,  and  the  change  of  air 
reaction  arising  in  this  way  is  evidently  dependent  upon  the 
velocity   of   rotation.     It   is   possible    to    analyze   with   some 
approximations  these  various  agencies  that  are  at  work,  and 
obtain  a  differential  equation  from  which  we  can  draw  con- 
clusions as  to  the  dynamic  stability.!    We  shall  not  do  this, 
however,  for  pitching  is  unavoidably  connected  with  a  change 
in  the  motion  of  the  center  of  gravity,  and  we  defer  the  whole 
question  to  the  more  accurate  and  complete  discussion  in  the 
next  chapter. 

STABILITY  IN  ROLLING 

96.  Suppose  the  machine  possessed  a  single  sustaining  mem- 
ber whose  leading  and  trailing  edges  were  straight  lines,  that  its 
body  were  that  of  a  solid  of  revolution,  that  its  tail  plane  were 
in  the  continuation  of  the  axis  of  the  body,  that  landing  gear 
struts,    wires,    etc.,    were    non-existent.     Suppose    that   while 
in  rectilinear  flight  with  the  axis  of  its  body  horizontal,  it  were 
made  to  roll  about  that  axis.     That  half  of  the  wing  whose  motion 
was  downwards,  would  have  its  angle  of  attack  increased;    the 
half  of  the  wing  moving  upwards,  would  have  its  angle  of  attack 

*  For  a  further  discussion  of  this  see  Devillers,  Chapter  XIII. 
t  Devillers,  Chapter  XIII. 


STABILITY   AND   CONTROLLABILITY  103 

decreased.  Consequently  there  would  be  a  damping  of  the 
motion,  dependent  upon  the  velocity  of  rotation.  Tail  plane 
and  rudder  would  also  assist  in  this.  When  the  motion  had 
died  out  there  would,  however,  be  no  restoring  couple;  that  is, 
there  is  no  static  couple  tending  to  restore  the  machine.  It  is 
obvious  that  during  the  motion,  and  in  the  displaced  position, 
with  the  wings  no  longer  horizontal,  the  vertical  component 
of  the  lift  no  longer  has  its  original  value,  and  unless  the  proper 
variation  of  speed  accompanied  the  process  the  altitude  of 
the  machine  would  change.  But  we  are  not  here  concerned 
with  all  the  complications  that  arise.  We  merely  are  interested 
in  seeing  that  there  is  no  restoring  couple. 

97.  While  the  discussion  that  has  been  given  does  not 
apply  in  toto  to  an  actual  machine, 
the  general  characterization  can 
be  carried  over,  and  we  see  that 
we  must  provide  a  means  of  creat- 
ing a  restoring  couple. 

One  way  of  causing  stability 
in  rolling  is  to  set  the  wings  at 
an  angle,  as  shown  in  the  figure. 
They  are  then  said  to  possess  a 

dihedral.  The  complete  action  of  this  is  difficult  to  trace, 
but  we  can  note  one  effect.  For  equilibrium  we  would  have 

zRv  cos  a  =  W. 

Now  let  the  machine  rotate  through  an  angle  6  so  as  to  lower 
the  left  wing.  The  vertical  component  of  the  air  reaction 
would  be,  assuming  R  has  not  materially  changed, 

Rv  COS  (a  —  0)  +Ry  COS  («-f-  0)  =  iRy  COS  a  -  COS  0. 

This  is  smaller  than  it  was  before.  Such  a  rotation  would 
then  be  accompanied  by  a  downward  motion  of  the  machine. 
This  would  increase  the  lift  on  the  left  (lower)  wing,  more  than 
on  the  right,  and  consequently  a  restoring  moment  would  be 
created,  that  would  make  the  machine  roll  back  to  its  original 
position.  The  dihedral  must  not  be  too  pronounced,  or  diffi- 
culty will  arise  from  the  lateral  effect  of  a  wind. 


104  THE  DYNAMICS  OF  THE  AIRPLANE 

The  fact  that  the  dihedral  produced  stability  through  the 
consideration  that  rolling  will  produce  a  downward  motion 
shows  how  impossible  it  is  to  separate  completely  different 
types  of  motion  in  the  discussion  of  stability.  They  are  all 
closely  connected,  and  rotation  about  one  line  will  cause  rota- 
tions about  other  lines,  though  perhaps  of  a  less  pronounced 
nature. 

98.  Controllability  in  the  lateral  sense  is  furnished  by 
means  of  the  ailerons.  Rectangular  portions  are  cut  from  the 
corners  on  the  trailing  edge  of  the  wings,  and  are  then  pivoted 
along  their  front  edges.  They  are  so  fastened  to  the  controls 


FIG.  47. 

that  the  raising  of  the  ailerons  on  one  side  is  accompanied  by  a 
lowering  of  those  on  the  other.  Their  action  is  obvious.  They 
merely  increase  the  lift  (and  the  drag)  on  one  side  and  decrease 
it  on  the  other.  This  gives  the  pilot  the  power  of  creating  at 
will  a  moment  that  tends  to  roll  the  machine. 

LATERAL  STABILITY 

99.  Lateral  stability  is  secured  by  means  of  a  fin,  in  the 
plane  of  symmetry,  to  the  rear  of  the  center  of  gravity.  The 
general  manner  in  which  it  functions  is  obvious.  In  case  the 
machine  is  made  to  yaw,  a  moment  is  created  tending  to  restore 
the  machine.  Of  course,  all  parts  of  the  machine  have  an 
effect  in  producing  the  restoring  couple.  Those  parts  well 
forward,  such  as  the  housing  of  the  motor,  tend  to  aggravate 
the  yawing. 

Controllability  in  direction  is  secured  by  means  of  a  rudder, 
usually  placed  at  the  rear  of  the  fin.  In  order  to  decrease 
the  muscular  exertion  that  the  pilot  must  use  to  turn  the 
rudder,  it  is  generally  balanced,  as  was  explained  in  the  case 
of  the  elevator.  Here  again  it  is  necessary  to  be  assured  that 


STABILITY  AND   CONTROLLABILITY  105 

the  force  on  the  rudder  will  not  pass  through  the  axis,  as  in 
this  case  the  pilot  would  not  be  sensible  of  any  pressure  on  the 
controls.  When  the  rudder  is  in  the  neutral  position,  it  acts 
as  a  portion  of  the  fin. 

100.  The  complete  analysis  of  the  action  of  the  rudder  is 
difficult,  for  its  turning  introduces  a  sequence  of  phenomena 
hard  to  follow.     The  case  is  much  more  complicated  than  that 
of  a  ship,  where  the  sustentation  is  in  no  way  dependent  upon 
speed. 

Imagine  that  the  rudder  has  been  turned  to  the  right.  A 
force  /  arises  on  the  rudder,  producing  a  moment  tending  to 
make  the  machine  yaw  to  the  right.  To  find  the  effect  on  the 
motion  of  the  center  of  gravity  we  must  apply  this  force  at 
that  point.  We  see  then  that  the  center  <  /  ^~v 

of  gravity  moves  slightly  to  the   left  at      I—  A 

first.  But  the  yawing  produces  a  force  F 
on  the  opposite  face  of  the  fin.  The  effect 
of  this  transferred  to  the  center  of  grav- 
ity will  be  to  make  the  machine  turn  to 
the  right.  The  total  force  tending  to 
make  the  center  of  gravity  describe  a 
curve  to  the  right  is  approximately  F—f. 
The  total  moment  tending  to  produce  rotation  about  a  vertical 
axis  is  the  resultant  of  the  moments  produced  by  the  rudder 
and  fin.  It  will  be  seen  that  a  large  fin  well  forward  will  make 
possible  a  considerable  turning  force  by  the  use  of  a  fairly 
small  rudder. 

The  combined  use  of  ailerons  and  rudder  in  turning  has 
already  been  discussed  in  §  53. 

101.  It  is  not  possible  to  separate  a  tendency  to  yaw  from  a 
tendency  to  roll,  for  the  two  will  occur  together.     Thus,  if  the 
machine  should  start  to  yaw  to  the  right,  the  left  side  of  the  wing 
will  have  a  greater  velocity  than  the  right,  and  consequently 
a  greater  lift,  so  that  the  machine  will  start  to  roll. 

The  fin,  while  designed  primarily  for  producing  stability  in 
direction,  also  contributes  to  stability  in  rolling,  if  properly 
placed.  Suppose  the  machine  had  no  dihedral,  and  that  it 


106  THE  DYNAMICS  OF  THE  AIRPLANE 

started  to  roll  in  the  direction  that  causes  the  left  wing  to 
lower.  The  lift  is  no  longer  vertical,  so  the  machine  will  start 
downwards.  Suppose  the  machine  had  a  vertical  fin  placed 
above  the  center  of  gravity.  It  will  have  been  displaced  from 
the  vertical  plane  in  the  rolling,  and  when  a  tendency  to  settle 
begins,  it  is  seen  that  a  force  arises  on  the  fin  that  tends  to  make 
the  machine  roll  back  toward  its  original  position.  If  the  fin 
had  been  below  the  center  of  gravity,  the  effect  on  it  would  be 
to  accentuate  the  roll.  The  efficiency  of  the  fin  in  creating 
stability  in  rolling  will  also  depend  on  its  distance  to  the  front 
or  rear  of  the  center  of  gravity. 

In  a  similar  way  the  dihedral  of  a  machine  adds  to  direc- 
tional stability.  For  if  the  machine  starts  to  yaw  to  the  right, 
the  effective  angle  of  attack  of  the  left  wing  is  increased,  that 
of  the  right  wing  decreased,  and  therefore  a  righting  moment 
arises. 

From  the  two  properties  that  have  been  deduced*  it  appears 
that  a  dihedral  in  the  wings  is  equivalent  to  a  certain  fin,  and 
can  for  practical  purposes  be  so  regarded. 

102.  Spiral  Instability. — A  type  of  instability  that  is  likely 
to  occur,  and  is  apt  to  prove  very  dangerous,  is  called  spiral 
instability.  It  can  be  caused  by  large  fin  surfaces  too  far  to 
the  rear  of  the  center  of  gravity.  Suppose  a  machine  with 
such  a  fin  were  banked  so  as  to  turn  to  the  left.  As  the  turn 
commences  a  pressure  arises  on  the  left  side  of  the  fin.  This 
causes  the  machine  to  rotate  so  as  to  keep  its  axis  along  the 
path.  The  outside  wing,  moving  faster  than  the  inside,  the 
lift  is  greater  there,  and  this  tends  to  increase  the  banking. 
Along  with  this  goes  a  slipping  towards  the  inside ;  for  a  shifting 
in  the  direction  of  the  lift  decreases  the  vertical  force  on  the 
machine.  This  slipping  causes  a  force  on  the  fin,  that  tends 
to  make  the  movement  more  pronounced.  As  a  result  of  this 
sequence  of  phenomena  the  machine  gets  into  a  spin,  or  rapid 
nose  dive,  from  which  the  pilot  may  be  unable  to  extricate  it. 


CHAPTER  VIII 

STABILITY—  (Continued) 

103.  A  GENERAL  discussion  of  the  question  of  stability  of 
an  airplane  can  be  made  by  the  methods  developed  by  Routh 
in  his  prize  essay  on  the  stability  of  a  dynamical  system.* 
The  application  of  these  methods    to  aviation  was   first  given 
by  Bryan,  f     His  results  are  general,  and  the  question  of  the 
stability  of  a  machine  for  any  slight  disturbance  from  a  position 
of  equilibrium  depends  upon  the  nature  of  the  roots  of  tv/o 
biquadratic    equations.     The    coefficients    in   these    equations 
depend  upon  the  construction  of  the  machine.     All  elements 
that  enter  into  the  machine's  construction,  the  wings,  the  stabil- 
izing surfaces,  the  controlling  surfaces,  contribute  to  the  value 
of  the  coefficients.     In  the  application  of  the  methods,  difficulties 
arise  in  the  determination  of  these  coefficients.     Bryan  himself 
applies  the  method  to  machines  of  certain  general  characteristics, 
for  which  he  can  obtain  with  considerable  certainty,   values 
for  the  coefficients  in  terms  of  wing  area,  angle  of  attack,  etc. 
Extensions  and  application  of  the  method,  by  obtaining  the 
values  of  the  coefficients  in  the  biquadratics    through  experi- 
ments on  models  in  wind  tunnels,  has  recently  been  made  by 
other  investigators,  t 

104.  Moving  Axes. — Many  problems  in  rigid  dynamics  are 
best  treated  by  means  of  moving  axes.     These  axes  are  fixed 

*  E.  J.  Routh,  "Stability  in  Motion,"  London,  Macmillan  Co.,  1877.  See 
also  his  "Advanced  Rigid  Dynamics,"  Chap.  VI. 

t  G.  H.  Bryan,  "Stability  in  Aviation,"  Macmillan  Co.,  1911. 

t  Bairstow,  "Technical  Report  of  the  (British)  Advisory  Committee  for 
Aeronautics,  for  1912-13,"  London,  Darling  &  Son.  In  this  report  Bairstow 
gives  simplifications  in  the  equations  of  Bryan,  and  develops  a  method  applicable 
to  a  machine  without  special  hypothesis  as  to  its  construction. 

107 


108  THE  DYNAMICS   OF   THE   AIRPLANE 

in  the  body,  move  along  with  it,  and  assume  continually 
changing  positions  and  directions.  They  can  afford  us  only 
indirectly  a  knowledge  of  how  far  the  body  has  moved,  but 
are  peculiarly  adapted  to  reveal  the  oscillations,  and  small 
changes  in  motion,  that  the  body  is  experiencing  at  any  instant. 
And  in  the  question  of  stability  it  is  exactly  such  points  that  are 
at  issue.  Use  of  moving  axes  was  first  made  by  Euler,  and 
equations  of  motion  for  such  axes  were  obtained  by  him.  Bryan, 
however,  found  the  particular  system  of  coordinates  used  by 
Euler  not  well  adapted  to  follow  the  motion  of  an  airplane, 
and  introduced  slight  changes  in  them. 

105.  Suppose  that  in  a  rigid  body  three  axes  be  chosen,  with 
their  origin  at  the  center  of  gravity.  As  the  body  moves, 
taking  this  system  of  axes  with  it,  we  fix  our  attention  upon 
it  at  some  particular  instant.  Its  center  of  gravity  has  a 
vector  velocity  V,  and  the  body  a  rotational  velocity  repre- 
sented by  another  vector  R.  Resolve  V  along  the  three  direc- 
tions in  space  that  are  occupied  at  that  instant  by  the  moving 
axes:  let  the  components  be  u,  v,  w.  Likewise  resolve  R: 
let  its  components  be  p,  q,  r.  We  thus  obtain  six  functions 
of  the  time,  and  the  equations  of  motion  will  be  obtained  by 
properly  expressing  these  six  functions  and  their  derivatives 
with  regard  to  the  time  in  terms  of  the  forces  and  moments 
acting  on  the  body.  Something  as  to  the  significance  of  the 
derivatives  can  be  obtained  by  considering  one  of  them,  for 
instance  du/dt.  It  measures  the  rate  at  which  the  component 
of  a  vector  along  a  varying  direction  is  changing.  It  is  there- 
fore not  a  component  of  the  acceleration  of  the  body,  for  a 
component  of  an  acceleration  is  the  rate  at  which  the  com- 
ponent of  the  velocity  along  some  fixed  direction  is  changing. 
Suppose  that  at  instant  /  the  origin  is  at  O,  and  let  the  position 
of  the  moving  axis  be  X.  Let  V  be  the  vector  velocity;  then  u 
is  the  projection  of  V  on  X.  At  a  subsequent  instant  let  the 
origin  be  0'  and  the  x  axis  be  denoted  by  X'}  and  the  velocity 
by  V'\  then  u'  is  the  projection  of  V  on  X'.  Thus  the  incre- 
ment of  u  is  u'  —u.  On  the  other  hand,  to  find  the  acceleration 
of  the  body  at  the  instant  /  we  must  project  V  upon  X,  instead 


STABILITY 


109 


of  on  the  new  position  of  that  axis.  The  acceleration  along  the 
fixed  direction  in  space,  which  at  any  instant  coincides  with  the 
position  of  the  moving  axis,  can,  however,  be  expressed  in 
terms  of  du/dt,  v  and  w.  Similar  remarks  apply  to  the  other 
quantities.  We  shall  not  derive  these  forms.  They  can  be 
found  treated  by  Routh.* 

106.  Choose  for  #-axis  a  line  in  the  plane  of    symmetry, 
for  instance,  parallel  to  the  propeller  axis,  directed  backward; 


FIG.  49. 

for  3/-axis,  a  line  perpendicular  to  the  plane  of  symmetry  to  the 
left  as  seen  by  the  pilot;  for  z-axis,  a  perpendicular  to  these, 
in  the  plane  of  symmetry,  directed  upwards.  The  axes  are 
further  so  chosen  that  the  origin  is  the  center  of  gravity  of  the 
machine. 

*  "  Elementary  Rigid  Dynamics,"  Chapter  V,  "Advanced  Rigid  Dynamics," 
Chapter  I. 


110  THE  DYNAMICS   OF  THE  AIRPLANE 

With  this  choice  of  axes  the  accelerations  of  the  center  of 
gravity  along  fixed  directions  in  space,  that  are  the  instan- 
taneous positions  of  the  moving  axes  are,  respectively: 

du  . 
—+wq-vr, 

dv  , 

—  +ur-wp, 

dw  .  * 

—  +vp-uq.* 

We  need  also  the  rates  of  change  of  the  angular  momenta 
about  the  various  axes.  We  denote  these  momenta  by  hi,  fe, 
hz'j  then 

hi=pA-qF-rE, 

h<2=qB-rD-pF, 
hz  =  rC-pE-qD, 

where  A  ,  B,  C  are  the  moments  of  inertia,  and  D,  E,  F  the  prod- 
ucts of  inertia.  The  machine  being  symmetric  we  have 
D  =  F  =  o.  The  rates  of  change  of  the  angular  momenta  are 
then: 


i     .  j        , 
—  -ph-3+rhi, 


107.  It  is  necessary  for  us  to  have  a  means  of  comparing 
the  orientation  of  the  machine  with  some  fixed  orientation. 
This  standard  of  reference  we  take  as  one  in  which  the  xy 

*When  comparison  is  made  with  Routh,  "Advanced  Rigid  Dynamics,"  §  5, 
it  will  be  found  that  the  signs  of  the  second  and  third  terms  are  changed.  This 
comes  from  the  fact  that  Routh  uses  a  right-hand  system  of  axes  while  we  have 
a  left-hand  system.  In  our  system  a  rotation  of  an  ordinary  screw  from  the 
#-axis  into  the  y-axis  would  advance  the  screw  along  the  negative  z-axis. 

f  Routh,  "Advanced  Rigid  Dynamics,  "  §  5. 


STABILITY 


111 


plane  is  horizontal.  From  this  position  rotate  the  machine 
first  about  the  z-axis  through  an  angle  $,  then  about  the  ;y-axis 
through  an  angle  6,  and  finally  about  the  #-axis  through  an 

*1 

angle  <f>.    A  rotation  about  the  y  \  axis  is  positive  when  it  turns 

z\ 


the  z 


x 


axis  into  the  x  \  axis.     The  original  positions  of  the  axes 


are  shown    by  the    letters  with  subscripts  o,  and  the    posi- 


yl  and  y< 


FIG.  50. 


tion  after  the  first  and  second  rotations  are  shown  respect- 
ively by  subscripts  i,  2,  and  the  final  positions  by  the 
letters  without  subscripts.  Further,  any  position  of  the  coor- 
dinate axes  can  be  obtained  by  a  unique  rotation  of  the  sort 
described.  Consider  a  position,  represented  by  x,  y,  z.  The 
angle  between  the  planes  xz0  and  XQZQ  determines  ^.  Let  the 
plane  XZQ  intersect  the  xoyo  plane  in  Ox\]  then  6  is  the  angle 
between  Ox  and  Oxi.  Let  Oyi  be  the  intersection  of  the  xoyo 
and  yz  planes;  then  <£  is  the  angle  between  Oy\  and  Oy.  To  fix 


112  THE  DYNAMICS   OF  THE  AIRPLANE 

the  orientation  of  the  machine  at  any  instant  we  draw  through 
the  center  of  gravity  axes  parallel  to  those  of  reference,  and 
determine  \f/,  6,  <f>  as  just  described. 

108.  Suppose  the  orientation  of  the  machine  is  continually 
changing;  then  $,  0,  <f>  are  functions  of  the  time.  From  their 
instantaneous  values,  and  those  of  their  derivatives  with 
regard  to  the  time,  we  can  obtain  the  instantaneous  values  of 
pj  q,  r.  We  shall  be  in  need  of  these  relations,  and  will  proceed 
to  their  determination. 

About  the  origin  in  Fig.  50  draw  a  sphere  of  unit  radius. 
Consider  the  point  C.  As  the  x,  y,  z  axes  move,  C  has  a  velocity 
dd/dt  along  the  arc  XZQ,  and  a  velocity  cos  6d\l//dt  normal  to  the 
plane  xOzo,  that  is,  along  xy.  Now  the  angular  velocity  r  is 
the  velocity  with  which  x  approaches  y,  that  is,  the  velocity 
along  xy.  But  xy  makes  an  angle  $  with  xy\.  Resolving 
in  the  direction  xy  the  velocity  dd/dt  along  ZQX,  and  cos  6d\l//dt 
along  xyi,  we  have 

r=  —sin  0-  —  hcos  <f>-cos6~. 
at  at 

Likewise  q  is  the  velocity  of  C  along  zx\  hence 

A  de  .  .  0  d$ 

0  =  cos  </>•-  —  hsm  0-cos  8~. 
at  at 

Finally,  r  is  the  velocity  of  z  away  from  y  along  yz.  But  the 
velocity  of  z  relative  to  Co  is  d<l>/dt,  and  that  of  Co  itself  is 
sin  6  -  d^/dt.  Hence  taking  account  of  the  positive  directions 
for  increasing  \J/  and  4>, 


When  the  angles  ^,  6,  4>  are  all  zero,  we  have 
d<f>  de  d\j/ 


*The  values   of  p,  q,   r  are  similar   to  the  well-known    Euler   geometrical 
equations.     See  Routh,  "Elementary  Rigid  Dynamics,"  §  256. 


STABILITY  113 

and  when  the  angles  are  small,  we  can  also  use  these  values 
for  p,  q,  r,  with  a  sufficient  degree  of  precision.  The  use 
of  the  exact  values  would  introduce  a  very  high  degree  of  com- 
plexity into  our  work. 

109.  Equations  of  Motion. — The  equations  of  motion  are 
easily  derived  from  the  expressions  that  have  been  given  in 
§  106.  The  motion  of  the  center  of  gravity  is  determined  by 
equating  the  expressions  for  the  linear  acceleration  to  the  forces 
per  unit  mass  acting  along  the  respective  axes.  The  rotation 
will  be  determined  by  equating  the  rates  of  change  of  angular 
momenta  to  the  moments  of  the  forces  about  the  respective 
axes. 

The  forces  acting  come  from  three  sources:  gravity,  pro- 
peller traction,  and  air  reaction.  The  components  along  the 
axes  of  the  force  of  gravity  depend  upon  the  orientation  of 
the  machine.  Let  ^,  6,  0  be  the  values  of  the  angles  giving 
the  orientation,  as  explained  in  §  107.  Then  the  components  of 
the  weight  per  unit  mass  are: 

g-sin0,         —  g-cos  0-sin  </>,         —  g-cos  0-cos  <£, 

along  the  x,  yt  z  axes,  respectively. 

The  moments  due  to  the  weight  are  zero. 

Next,  consider  the  traction  of  the  propeller.  Let  it  be 
parallel  to  the  x-axis,  of  numerical  value  H  per  unit  mass,  and 
applied  at  a  distance  h  above  the  x-axis.  Its  components  are: 

-H,        o,        o. 

The  moments  of  this  force  are: 

o,         -hH,        o,. 

about  the  x,  y,  z  axes,  respectively. 

There  remains  the  air  reaction.  It  depends  upon  the 
instantaneous  velocity,  translational  and  rotational;  that  is, 
upon  u,  v,  Wj  pj  q,  r.  No  simple  expression  can  be  given  for  it. 
We  denote  the  components  of  this  force,  per  unit  mass,  at  a 
given  instant  by  X,  F,  Z,  and  its  moments  about  the  respective 
axes  by  L,  M,  N. 


114  THE  DYNAMICS  OF  THE  AIRPLANE 

The  equations  of  motion  now  become 

—+wq-  w  =g-sin  e+X-H, 
at 

dv 

- — h  ur— wp  —  —  g-cos  0-sin  0+F, 

for  the  motion  of  the  center  of  gravity,  and  (by  making  use 
of  the  angular  momenta  in  terms  of  the  angular  velocities, 
remembering  that  the  products  of  inertia  D  and  F  are  zero), 


B 


for  the  rotation  about  the  axes. 

110.  We  shall  employ  the  equations  of  motion  to  determine 
the  effect  of  a  disturbance  that  a  machine  might  undergo  while 
it  is  in  a  state  of  steady  motion.  We  limit  ourselves  to  the 
simplest  case,  for  even  in  this  case  the  equations  with  which 
we  must  deal  assume  a  sufficiently  complicated  form.  The 
analysis  of  the  subject  which  has  been  developed  to  date,  will 
not  enable  us  to  treat  a  general  displacement  from  a  general 
position  of  equilibrium  in  which  a  machine  might  find  itself. 
We  can  study  some  simple  cases,  those  applying  to  normal 
states  of  flight.  If  we  find  in  these  instances  a  high  degree 
of  equilibrium,  we  will  be  led  to  infer  that  the  machine  can  be 
regarded  as  air-worthy,  and  will  furnish  a  means  of  transit  in 
general  of  a  sufficient  degree  of  security. 

Suppose  the  machine  is  in  rectilinear  horizontal  flight. 
The  xz  plane  is  vertical.  Assume  also  that  the  #-axis  is  hori- 
zontal. (We  have  assumed  the  #-axis  parallel  to  the  pro- 


STABILITY  115 

peller  axis.  The  new  assumption  is  really  not  a  restriction, 
as  the  equations  that  follow  could  easily  be  modified  in  such 
a  way  as  to  take  care  of  the  case  where  in  the  horizontal  rec- 
tilinear flight  the  £-axis  is  inclined  at  a  certain  angle.  There 
would,  in  general,  be  some  angle  of  attack  for  the  wings  for 
which  the  #-axis  is  horizontal.)  In  this  state  of  steady 
motion,  v,  w,  p,  q.  r  are  all  zero,  while  u  has  a  constant  value  U 
(which  is  negative,  on  account  of  the  direction  in  which  the 
#-axis  is  chosen).  The  steady  motion  is  represented  by  the 
equations: 

o  =  X0-H0, 

o=Y0, 

o  =  Z0, 
o  =  mLo, 


o  =  mNo, 

where  a  letter  with  subscript  zero  denotes  the  value  of  that 
quantity  for  the  state  of  steady  motion. 

Now  imagine  a  disturbance  to  take  place.  Its  exact  nature 
we  do  not  specify,  nor  do  we  know  exactly  what  effect  it  will 
instantaneously  have  upon  the  machine.  We  shall  merely 
say  that  it  has  altered  all  the  velocities  of  the  machine.  Thus 
v,  w,  p,  q,  r  have  values  other  than  zero,  while  u  takes  the  value 
U+u.  (The  change  in  the  significance  of  u  will  cause  us  no 
confusion.)  As  is  usual  in  the  case  of  such  discussions,  we 
assume  that  u,  v,  w,  p,  q,  r  are  small  enough  that  their  squares 
and  products  can  be  neglected. 

111.  The  orientation  of  the  machine  will  be  different.  We 
take  the  orientation  prior  to  the  disturbance  as  the  one  of 
reference.  After  the  disturbance  the  position  of  the  machine 
will  be  represented  by  ^;  6,  <£,  all  of  which  we  assume 
sufficiently  small  for  us  to  write 

d<l>  dB  dty 


116  THE   DYNAMICS   OF   THE   AIRPLANE 

The  disturbance  will  have  changed  all  the  components  of 
the  air  reaction.  Consider  one  of  them,  X.  Putting  in  evi- 
dence the  quantities  upon  which  it  depends  we  obtain 

X(U+u,  v,  w,  pi  q,  r)  =  X(Ut  o,  o,  o,  o,  o) 

M+?M+^.          ' 

dp         dq         dr 
That  is, 

X  =  Xo+uXu+vX9+wXu+pX9+qXq+rXr, 

where  XQ  is  the  value  of  X  prior  to  the  disturbance,  and  Xu 
Xv,  .  .  .  Xr  are  constants  called  "  resistance  derivatives,"  by 
Bryan.  In  all,  we  shall  have  thirty-six  resistance  derivatives, 
namely: 

5,,  S  =  X,  Y,  Z,  L,  M,  N, 

s  =  u,  v,  wy  p,  q,  r. 

On  account  of  the  symmetry  of  the  machine  eighteen  of  the 
resistance  derivatives  vanish.  Those  that  vanish  are 


s  =  v,p,r, 
and 

&,          S=Y,L,N, 


To  show  that  these  eighteen  resistance  derivatives  are  zero, 
consider  one  of  them,  for  instance,  Xv.  If  it  were  not  zero, 
a  sideways  velocity  v,  would  cause  an  increase  vXv  in  the  force 
along  the  s-axis,  and  this  would  be  positive  or  negative  accord- 
ing as  the  displacement  were  towards  the  left  or  right,  which 
could  not  be  the  case  if  the  machine  were  symmetrical. 

Consider  also  Yq.  If  it  were  different  from  zero,  a  dip 
in  the  machine  would  produce  a  sideways  force  of  a  different 
direction  from  that  produced  by  a  tip. 

Finally,  let  H0+5H  be  the  new  propeller  traction. 


STABILITY  117 

The  differential  equations  that  represent  the  motion  sub- 
sequent to  the  disturbance  will  be  obtained  by  substituting  the 
new  values  of  the  velocities  in  the  six  equations  at  the  end  of 
§  109,  dropping  products  of  small  quantities,  using  the  new 
values  for  the  various  forces  and  moments,  and  at  the  same 
time  making  use  of  the  relations  for  steady  motion. 

We  obtain  thus: 


~ 

d*w 
— 


B  -2  =  m(uMu+wMw+qMq—h>  5H), 


at         at 

These  six  equations  divide  into  two  groups.  The  first, 
third,  and  fifth  contain  only  u.  w,  q,  and  their  derivatives;  the 
second,  fourth  and  sixth  contain  v,  p,  r  and  their  derivatives. 
The  first  group  determines  motion  in  the  xy  place.  Such 
motions  are  called  symmetric  or  longitudinal  oscillations.  The 
second  group  determines  oscillations  that  are  called  by  Bryan 
the  asymmetric  oscillations. 

112.  As  the  equations  contain  also  0  and  0,  we  replace 
p  and  q  by  d<f>/dt  and  dO/dt,  respectively,  and  thus  take  u,  vt 
Wj  <£,  0,  r  as  the  quantities  to  be  sought. 

Finally,  we  take  6/7  =  o;  that  is,  we  assume  that  the  rota- 
tion of  the  propeller  changes  in  such  a  way  that  the  traction  is 
unaltered  by  the  disturbance.* 

*  For  discussion  of  the  case  where  this  assumption  is  not  made  see  Bryan, 
loc.  cit,  p.  28. 


118  THE  DYNAMICS   OF   THE   AIRPLANE 

We  make  these  substitutions,  represent  differentiation  by 
the  symbol  D,  and  divide  the  equations  into  the  two  groups 
mentioned.  For  the  longitudinal  motion  we  have: 

(D-Xu)u-Xww-(XgD+g)=o, 


where  we  have  put  kB2  =  B/m;   and  for  the  asymmetric  oscilla- 
tions we  have: 

(D-Yv)v-(YpD-g)<i>  +  (U-Yr)r  =  o, 

A2D2-LpD)<}>-(kE2D+Lr)r  =  o, 


where  kA2  =  A/m}  kE2  =  E/m,  kc2  =  C/m. 

These  are  the  fundamental  equations  with  which  we  have 
to  deal.  If  we  had  assumed  that  the  steady  motion  had  been 
along  a  line  making  a  constant  angle  with  the  horizon,  slightly 
different  equations  would  be  obtained.* 

113.  The  equations  that  we  have  to  solve  are  linear  equa- 
tions with  constant  coefficients.  The  solution  is  most  readily 
accomplished  by  means  of  symbolic  operators,  f  Consider  first 
the  equations  for  longitudinal  motion.  Let 

-Xu,  -Xw  -(XaD+g) 
-Zu,  D-ZW,  -(ZQ+U)D 
-Mu,  -Mw,  (kB2D2-MqD) 

The  determinant  on  the  right  is  to  be  developed  according  to 
the  ordinary  method,  the  symbol  D  representing  differentiation. 
When  this  is  done  we  find 


*  Bairstow,  loc.  cit.,  Bryan,  Chapter  I,  Cowley  and  Evans,  Chapter  XI. 
t  Murray's    "  Differential    Equations,"    Chapter   XI.    Wilson's    "Advanced 
Calculus,"  p.  223. 


STABILITY  119 

where  A,  B,  C,  D,  E  are  constants,  whose  explicit  values  will 
be  given  presently.  The  quantities  «.  w,  6  are  then  all  solu- 
tions of  the  single  equation  of  the  fourth  order, 


In  a  similar  way  we  put 

A'-ir     rr       -(YpD-g),  U-Yr, 

-_.,.     (kA2D2-LPD)          -(kE2D+Lr)  , 
-N,,     -(kE2D2+NPD),       (kc2D-Nr) 
which  gives,  upon  development, 


The  quantities  v,  0,  r  are  then  all  solutions  of 


The  conditions  for  stability  can  now  be  given  in  a  general 
way.  The  quantities  u,  w,  0  will  be  linear  combinations  of 
the  form 


where  Xi,  X?,  Xa,  X4  are  the  four  roots  of  the    quartic 


and  v,  </>,  r  will  be  of  the  same  form,  where  Xi,  X2,  Xs,  X4  are 
roots  of 


«  In  order  that  the  machine  be  stable  it  is  therefore  necessary 
that  the  roots  of  the  two  quartic  equations  have  their  real  parts 
negative.  For  if  this  condition  is  fulfilled  the  values  of  u, 
v,  w.  9,  0,  r  become  smaller  and  smaller  as  the  time  increases; 
they  thus  approach  zero,  and  a  steady  motion  is  therefore 
resumed.  If,  on  the  other  hand,  the  real  part  of  one  of  the 
roots  is  positive,  the  term  corresponding  to  it  increases  and 
instability  will  result.  It  is  to  be  noted  that  a  machine  can 
have  longitudinal  stability,  but  asymmetric  instability,  or 
vice  versa.  (We  shall,  however,  see  later  that  asymmetric 
stability  is  the  more  difficult  to  obtain.) 


120 


THE   DYNAMICS   OF   THE   AIRPLANE 


114.  The  necessary  condition  that  the  real  roots  be  negative, 
and  the  real  parts  of  the  imaginary  roots  be  negative  in  a 
quartic  equation  is  given  by  Routh.*  Let  the  equation  be 


Then  the  roots  will  have  the  property  mentioned  if,  and  only 
if,  the  coefficients 

a,  b,  c,  d,  e, 

and  the  discriminant,  called  Routh's  discriminant, 

bcd-ad2-eb2 
are  all  positive, 

An  airplane  will  therefore  be  stable  longitudinally  if 

A,  B,  C,  D,  E,     and    BCD-AD2-EB2 

are  all  positive. 

It  will  be  asymmetrically  stable  if 

A,,  Blt  Ci,  Di,  £1,     and     B.C^-A^-E^ 

are  all  positive. 

115.  If  the  determinant  A  be  developed,  we  find: 

A  =  kJ,  B=-(Mq+XukB2+Zwk2), 


zw. 


Mw, 


XV)  Xw 
7      7 

Ss  U .       / s  ir 


D=- 


fu,  MWJ  Mq 
E=-g   ZM,   Zw     . 


In  the  same  way,  by  developing  A'  we  obtain: 


^i  = 


k2 


*  "Stability  in  Motion,"  p.  14. 


STABILITY 

-kj,  -kj 

— 

kE 

2,    Lr 

-LP,    - 

-kF?    , 

he2,      kc2 

2,    Nr 

NP, 

kc2 

'.,  Yp,      o 

-   F.,  o, 

Yr-U    - 

\-    Lp,     Lr 

c,     Lpj      —  KE 

Lv,    -kA2, 

Lr 

NP)    Nr 

r,,  NP)      kE2 

NV)  kE2, 

Nr 

V      V       V  —77 

JL  V)     *  p,     *  r       U 

+g 

L,,    -kE2    , 

Lc,     Lp.     Lr 

N0, 

kc2 

121 


*,     Lr       . 


116.  The  whole  question  of  determining  the  stability  of  a 
machine  depends  upon  the  determination  of  the  resistance 
derivatives,  the  knowledge  of  which  will  give  the  coefficients 
in  the  two  quartics.  This  is  a  practical  question  which  is 
solved  by  experiments  on  a  model  in  a  wind  tunnel.  We  shall 
give  only  a  slight  indication  of  the  procedure.  The  question 
is  discussed  at  length  by  Bairstow.* 

Consider  the  resistance  derivatives  that  depend  on  u. 
By  means  of  instruments  which  hold  the  model  in  the  wind 
tunnel,  the  resistance  XQ  for  the  velocity  U  is  determined,  the 
#-axis  being  in  the  direction  of  the  wind.  We  assume 

X  =  cn\ 
where  c  is  a  constant  and  u  the  velocity.     Hence 

(}X -yr  XQ XQ 

==  .A. «  ==  2  CU  ==  2  U  '      ~  =:  2          . 


*  Loc.  cit.,  pp.  154-158.  See  also:  Hunsaker,  "Experimental  Analysis  of 
Inherent  Longitudinal  Stability  for  a  Typical  Biplane."  First  Annual  Report 
of  the  (American)  National  Advisory  Committee  for  Aeronautics,  Hunsaker, 
"Dynamical  Stability  of  Aeroplanes,"  Smithsonian  Miscellaneous  Collection, 
Vol.  62,  No.  5.  These  two  papers  by  Hunsaker  will  be  referred  to  as  Hunsaker 
(i),  Hunsaker  (2),  respectively.  The  second  paper  especially  gives  a  full  dis- 
cussion of  the  subject  referred  to  here. 


122  THE   DYNAMICS   OF   THE   AIRPLANE 

In  the  same  way  Z«,  Af«,  Nu  are  found  from  a  knowledge  of 

ZQ,  MQ. 

The  derivatives  that  depend  upon  v  are  found  by  turning 
the  model  about  the  s-axis  through  some  fixed  angle,  for  the 
effect  of  such  a  turning  is  to  give  a  component  of  the  air  reaction 
parallel  to  the  ^-axis.  In  the  same  way  the  derivatives  depend- 
ent on  w  are  found  by  making  measurements  after  the  machine 
is  turned  about  the  y-axis  through  some  angle. 

The  derivatives  that  depend  upon  p,  q,  r  are  termed  rotary 
derivatives  by  Bryan.  They  are  determined  by  oscillating  the 
machine  about  the  various  axes,  controlling  the  oscillations  by 
means  of  springs,  and  measuring  the  damping  by  means  of 
photography. 

A  discussion  of  the  range  in  numerical  value  that  may  be 
expected  in  the  resistance  derivatives  for  machines  generally 
in  use  will  also  be  found  in  the  report  given  by  Bairstow. 

117.  As  an  example  of  the  numerical  values,  we  take  those 
given  by  Bairstow  for  a  typical  machine.  They  are: 

kB2=     25  m=        40  slugs* 


Xu=- 

.14 

Zu=-  .80 

Mu  = 

0 

xw= 

.19 

Zw=  —2.89 

Mw  = 

2.66 

xq= 

•5 

Z3=          9 

Mq  = 

—   210 

kA2= 

25 

kc2=        35 

W* 

o 

Yv=- 

•25 

Lv=       .83 

Nv  = 

•54 

Yp  = 

i 

Lp=  —  200 

N>  = 

28 

Yr=~ 

3 

Lr=             65 

Nr  = 

•37 

The  equation  for  longitudinal  stability  is 


and  for  lateral  stability 

X4+9.3iX3+9.8iX2-f-io.i5X-.i6i=o. 

"The  engineering  unit  of  mass,  equal  to  32.16  pounds.     It  is  used  here  in 
order  to  have  consistent  units.     See  appendix  for  a  further  discussion. 


STABILITY  123 

The  Routh  discriminant  for  the  first  equation  is  found  to  be 
positive.  Therefore  the  machine  possesses  longitudinal  stability. 
The  negative  term  in  the  second  equation  shows  the  machine 
to  be  asymmetrically  unstable.  Disturbances  therefore  would 
need  to  be  corrected  by  the  assistance  of  the  controls.  Bairstow 
shows  that  the  machine  becomes  asymmetrically  stable  when 
gliding  at  a  glide  of  i  :6  with  the  propeller  cut  off.  To  prove  this 
it  is  merely  necessary  to  have  assumed  that  the  steady  motion 
had  been  in  a  line  inclined  at  some  angle  to  the  horizon.  The 
development  is  similar  to  that  which  has  been  given  here. 

118.  The  stability  of  a  machine  will  depend  upon  its  speed, 
and  a  machine  may  be  stable  at  one  speed  and  unstable  at  another. 
This  is  due  to  variation  of  the  various  resistance  derivatives. 
Thus,  while  it  is  possible  to  make  a  machine  with  a  considerable 
speed  range  through  a  change  of  elevator  setting,  the  machine 
cannot  be  expected  to  be  uniformly  stable  at  all  the  speeds 
at  which  it  may  fly.     At  high  speed  the  pilot  may  be  able  to 
abandon    his    controls,    the   machine   possessing   so    great    an 
inherent  stability  as  to  be  able  to  "fly  itself,"  while  at  low 
speeds  constant  watchfulness  and  attention  may  be  necessary. 
An  interesting  example  of  this  sort  is  given  by  Hunsaker.*     In 
his  report  the  longitudinal  stability  of  a  machine  is  investigated 
for  speeds  varying  from  79  miles  an  hour,  corresponding  to  an 
angle  of  attack  of  i°,  to  a  speed  of  43.7  miles  an  hour,  corre- 
sponding to  an  angle  of  attack  of  15°. 5.     Instability  at  low 
speeds  corresponding  to  an  angle  of  attack  larger  than  io°.5 
results  from  the  Routh  discriminant  becoming  negative.! 

119.  The  general  discussion  of  the  quartic  equations  under 
consideration   presents    considerable    difficulty.     Bairstow   has 
given  an  approximate  factorization  of  the  equations.!     That  is, 
he  has  given  factors  that,  while  not  algebraically  exact,  give 
a  fairly  close  approximation  when  the  numerical  values  of  the 
coefficients  are  used.     And  from  the  factors  he  gives  a  general 

*  (i)  Loc.  cit.,  pp.  47-51. 

t  Hunsaker  (2),  p.  65,  considers  the  lateral  stability  of  a  Clark  tractor  and 
shows  it  is  laterally  stable  except  at  low  speed. 
I  Loc.  cit. 


124  THE   DYNAMICS   OF  THE  AIRPLANE 

idea  of  the  properties  of  the  plane  that  contribute  to  its  sta- 
bility. 

The  equation  for  longitudinal  stability  Bairstow  factors  into 


In  general,  it  may  be  said  that  the  first  factor  represents  a 
very  short  oscillation,  which  in  the  majority  of  machines  will 
die  out  quite  rapidly.  The  second  factor,  however,  represents 
a  relatively  long  oscillation,  which  is  dependent  in  its  nature 
upon  the  speed,  and  may  cause  instability  at  the  low  speeds. 
The  factored  form  for  the  equation  for  lateral  stability  is 


It  can  be  shown  in  general  that  the  real  root  corresponding 
to  the  second  factor  is  negative,  and  that  the  real  parts  of 
those  corresponding  to  the  third  factor  are  negative.  Thus 
these  factors  indicate  stability.  A  consideration  of  the  first 
factor  shows  that  stability  requires  that  EI  and  DI  be  of  the 
same  sign.  It  is  found  that  DI  can  be  universally  regarded  as 
being  positive.  Thus  EI  must  be  rendered  positive.  This 
condition  is  the  most  difficult  to  obtain  in  construction.* 

120.  As  an  example  of  the  accuracy  of  the  approximate 
factorization   we   shall   consider  one  of   the   cases   considered 
by  Hunsaker.f    The  data  obtained  by  experiments  are: 
i  =  angle  of  attack  =  i°. 

Velocity  =  79  miles  an  hour,    U=—  115.5   foot-seconds. 

7^  =  55.9  slugs,  KB  =  34- 

Xu=—.i2&,        Xw—       .162,         Mw—     1.74 


This  gives  4  =  34,  £=289,  C  =  834,  £=115,  E  =  3i,  BCD-AD* 
iSXio6.    Hence  the  machine  is  longitudinally  stable. 


*  For  further  discussions  of  the  influence  of  design,  and  general  consideration 
of  this  question,  see  Bairstow,  loc.  cit.  Hunsaker  (i),  (2.)  Cowley  and  Evans, 
Chap.  XII. 

t  (i).  Article  13,  Case  I. 


STABILITY  125 

The  equation  for  longitudinal  oscillations  is 


The  approximate  factorization  from  Bairstow's  formulae  is 

(X2+8.5X+24.5)(X2+.i25X+.o374)=o. 
The  roots  obtained  from  the  first  factor  are 


27T 

which  give  the  short  oscillation  of  period  --  =  2.5  seconds. 

2.54 

The  roots  obtained  from  the  second  factor  are 
X=  -.063  ±.183*, 

which  give  the  long  oscillations  of  period  34.3  seconds. 

More  accurate  values  of  the  roots  of  the  quartic  for  the 
same  machine  are  given  by  Wilson,*  and  are 

X=  —4.180  ±2.4302', 
X=  —   .0654  ±   .1872. 

121.  Effect  of  Gusts.  —  We  have  so  far  considered  the  dis- 
turbance occurring  while  the  machine  is  in  still  air.  As  the 
atmosphere  is  in  continual  and  irregular  movement  the  effect 
of  gusts  must  be  considered.  A  rather  comprehensive  study  of 
this  nature  has  been  made  by  Wilson.*  The  actual  nature  of 
gusts  that  occur  in  the  air  can  obviously  not  be  represented  by 
mathematical  means,  because  of  our  lack  of  knowledge  of  the 
exact  conditions  and  fluctuations  in  the  atmosphere.  How- 
ever, we  can  assume  certain  "  mathematical  gusts  "  which 
would  seem  to  have  as  unstabilizing  an  effect  on  an  airplane 
as  an  actual  gust,  and  by  becoming  assured  of  a  satisfactory 
behavior  of  the  machine  in  such  mathematical  gusts  of  widely 
different  nature,  we  can  regard  the  machine  as  a  vehicle  possess- 
ing an  air-worthiness  sufficient  for  the  conditions  which  it  may 

*  E.  B.  Wilson,  "Theory  of  an  Aeroplane  Encountering  Gusts."  First  Annual 
Report  of  the  (American)  Advisory  Committee  on  Aeronautics,  Report  No.  i, 
part  2,  p.  58.  The  quartic  given  by  Wilson  is 

34\4+288.7X3+833.oX2+ii5.iX+3i.i8  =  o. 


126  THE  DYNAMICS  OF  THE   AIRPLANE 

in  practice  be  expected  to  encounter.  The  equations  that  have 
been  developed  in  what  precedes  give  the  basis  for  an  investiga- 
tion of  the  nature  mentioned. 

We  shall  merely  indicate  the  method,  without  going  into 
the  details  of  the  development,  for  considerable  calculation  is 
involved.  Reference  can  be  made  to  the  detailed  and  clear 
report  by  Wilson. 

122.  Suppose  first  that  the  machine  is  flying  horizontally 
and  encounters  a  gust  moving  parallel  to  the  x-axis.  We  think 
of  it  as  being  caused  by  a  motion  of  the  air  parallel  to  the  £-axis, 
with  veolocity  «i,  which  we  shall  take  to  be  positive  when 
along  the  negative  #-axis.  A  displacement  of  the  machine 
results,  and  the  gust  is  no  longer  directly  along  the  rr-axis. 
But  assuming  that  all  displacements  are  relatively  small,  we 
shall  say  that  the  gust  continues  unchanged  along  the  #-axis. 
The  machine  has  an  altered  velocity  along  the  s-axis  itself,  of 
amount  u,  according  to  our  former  notation.  Therefore  the 
total  change  in  relative  wind  is  u-\-u\.  This  is  consequently 
the  quantity  by  which  we  should  multiply  the  resistance 
derivative  Xu  in  the  equations  of  motion. 

In  the  general  case  we  shall  assume  that  the  gust  has  com- 
ponents ui,  vij  wi}  pij  qij  r\.  The  equations  for  the  longi- 
tudinal motion  will  then  be  obtained  by  altering,  as  indicated 
above,  the  quantities  by  which  the  various  resistance  deriv- 
atives are  multiplied.  We  have  then  for  the  equations  of  motion: 


-MuU-Mww+(kB2D2-MqD)d  =  Muui  +Mwu>i  +Mqqi. 

In  these  equations,  u\,  vi,  .  .  .  .  r\  are  supposed  to  be  known 
functions  of  the  time  which  vanish  for  the  moment  when  the 
gust  commences.  The  solution  will  consist  of  the  complimen- 
tary function  and  the  particular  integrals.  The  first  step  is  to 
solve  the  system  of  equations  algebraically.  We  shall  take 
the  result  for  u  as  typical.  We  find 

Aw 


STABILITY  127 

where  A  is  the  determinant  used  before,  and  AI,  A2,  AS,  are 
determinants  obtained  from  A  by  replacing  the  elements  of  the 
first  column  by  certain  of  the  resistance  derivatives.  These 
determinants  contain  the  operator  Z>;  they  are  to  be  expanded 
as  though  D  were  an  algebraic  quantity,  and  then  applied  as 
operators  to  the  known  functions  «i,  w\,  q\.  The  form  of  the 
determinants  makes  a  general  development  very  difficult. 
For  a  particular  machine  the  knowledge  of  the  resistance 
derivatives  allow  the  operators  AI,  A2,  AS  to  be  completely 
determined.  We  can  then  write  as  a  final  equation, 


where  u(t)  is  a  known  function  of  t  as  soon  as  we  have  assumed 
a  character  for  the  gust.  Similar  equations  exist  for  w  and  0. 
Represent  the  particular  integrals  by  7W,  Iw,  I9,  respectively. 
We  then  have  for  the  final  solutions  of  the  equations: 


where  Xi,  \2,  Xa,  X4  are  the  roots  of  the  quartic  that  gives  the 
longitudinal  motion,  and  the  quantities  Cy  are  constants.  These 
constants  must  be  determined  so  that  u,  w,  6  all  vanish  for 
/  =  o,  the  time  at  which  the  gust  commences.  The  fact  that 
the  expressions  obtained  on  dropping  /„,  IW}  I0)  must  satisfy 
the  differential  equations  enables  us  in  the  first  place  to  deter- 
mine the  ratios  en  :  c2i  :  c3i  (i=i,  2,  3,  4).  The  solutions 
can  then  be  expressed  in  terms  of  en,  Ci2,  c^,  c^.  The 
values  of  these  are  to  be  determined  so  that  u,  w,  0,  q  =  dB/dt 
are  all  zero  for  t  =  o.  This  gives  us  four  equations  involving 
the  constants  and  the  four  quantites  7«o,  7^,  7e0,  I'M,  which 
are  the  values  of  the  particular  integrals  for  /  =  o.  While 
imaginary  quantities  occur  in  the  process  of  the  work,  the 
final  result  can  be  put  in  a  trigonometric  form  free  from  imag- 
inaries.  The  details  of  this  development  are  given  in  Wilson's 
report,  articles  2,  3,  4. 


128  THE   DYNAMICS   OF   THE   AIRPLANE 

123.  As  an  illustration  of  the  application  of  the  method 
we  shall  take  the  first  gust  treated  by  Wilson,  for  the  machine 
referred  to  in  §  120.  Let  the  gust  behead-on,  and  represented 
by 

Ui=J(i-e   -2<),         wi=qi=o. 

It  is  seen  that  the  gust  increases  slowly  from  o  to  /  in  intensity. 
The  values  of  the  particular  integrals  found  by  Wilson  are: 

-2<),  /.   /«o=-.  753-^ 

IM=  -.c82/, 

I0  •=  -  .00495/0  ~-2',  /flo  =  -  .0049/, 

//  =  .  00099/0  ~-2',  I<®'  =  -00099/. 

The  complete  solution  of  the  differential  equations  then  gives: 
w  =  /0--065%622  cos  .i87/4-.63o  sin  .1870 


;  =  /0~4-18'(  —  .004  cos  2.43/4-  .003  sin  2.43*) 
-/0--065%o78  cos  .187/4-  .059  sin  .187*)  +  .o82/0--2', 
=  /0~-0654'(.oo495  cos  .i87/  —  .0031  sin  .iSy/)  —  .00495/0"  '2t- 


The  effect  of  the  gust  is  given  by  Wilson  as  follows:  "  (i)  The 
machine  takes  up  an  easy  slowly  damped  oscillation  in  u  of 
amplitude  about  89  per  cent  of  /;  after  the  oscillation  dies 
out  the  machine  is  making  a  speed  /  less  relative  to  the  ground 
and  hence  the  original  relative  speed  to  the  wind.  (2)  There  is  a 
rapidly  damped  oscillation  in  w  of  rather  small  magnitude  and 
a  slowly  damped  one  of  about  10  per  cent  of  /,  the  final  con- 
dition being  that  of  horizontal  flight.  (3)  There  is  a  slow 
oscillation  in  pitch  of  about  .oo58/  radians,  or  about  -32/0. 
If  the  magnitude  of  /  is  great,  the  pitching  becomes  so  marked 
that  the  approximate  method  of  solution  can  no  longer  be 
considered  valid  —  a  gust  of  20  foot-seconds  causing  a  pitch 
of  some  6°.  As  the  period  is  long  (about  one-half  minute), 
the  pilot  should  have  ample  time  to  correct  the  trouble  before 
it  produces  serious  consequences." 


STABILITY  129 

124.  During  the  interval  that  elapses  between  the  com- 
mencement of  the  gust  and  the  acquirement  of  the  subsequent 
steady  motion  the  altitude  of  the  machine  changes.  It  is 
possible  to  determine  this  change.  Let  the  vertical  velocity 

be  represented  by  -~,  £  being  measured  upwards.     Then,  by 
dt 

resolving  u  and  w  along  the  vertical  we  have 

~  =  w  cos  6+(V+u)  sin  9. 
dt 

Hence,  approximately,  the  change  in  altitude  is 

$=  (  (w+Vti)dt. 
Jo 

For  the  machine  and  gust  considered  this  becomes 
'°65%5  cos  .I87/-.4  sin  . 


=  J  f 

Jo 


sin  .i8yO  +  2.  5^-^-3.  5]. 


When  the  steady  motion  has  been  reached  the  limit  of  this  is 
—  3.57.  For  a  head-on  gust  /  is  negative.  Hence  the  machine 
would  rise  70  feet  on  encountering  a  head-on  gust  of  magnitude 
20  foot-seconds. 


APPENDIX 


1.  In  the  fundamental  equation, 

F  =  kAV2, 

that  we  have  used  for  the  air  pressure,  the  value  of  the  constant 
k  depends  upon  the  units  we  are  using.  Thus  for  pressure  on 
a  flat  plane,  we  have: 

F  =  .oo$oAV2,  if  A  is  in  sq.  ft.,  V  in  mi.  per  hr.,  F  in  Ibs. 
F  =  .  00143^4  F2,  if  A  is  in  sq.  ft.,  V  in  ft.  per  sec.,  F  in  Ibs. 
F=.  075^4  F2,  if  A  is  in  sq.  m.,  V  in  m.  per  sec.,  F  in  kg. 
F  =  .0057^4  F2,  if  A  is  in  sq.  m.,  F  in  km.  per  hr.,  F  in  kg. 

In  case  we  wish  to  compare  the  results  that  different 
experimenters  obtain  for  certain  constants,  it  is  necessary  to 
properly  take  account  of  the  units  employed,  before  an  opinion 
can  be  formed  as  to  the  agreement  of  the  results.  Similarly, 
if  we  wish  to  adapt  to  English  units  the  lift  and  drag  coefficients 
for  a  wing  which  has  been  tested  in  a  wind  tunnel  where  metric 
units  are  employed,  it  is  necessary  to  have  a  ready  means  of 
making  the  required  transformations. 

2.  Let  us  write 


where  p  denotes  the  density  of  the  air.  The  dimensions  of  p, 
A,  V2  are  respectively,  ML~3,  L2,  L2T~2.  Therefore  the  dimen- 
sions of  p^4F2  are  MLT~2.  But  the  dimensions  of  force,  F} 
are  also  MLT~2.  Consequently  the  constant  C  is  dimensionless, 
that  is,  it  is  an  absolute  constant,  independent  of  the  units 

131 


132  APPENDIX 

employed,  provided  the  system  of  units  is  self  consistent,  that 
is,  the  unit  of  force  is  the  force  required  to  give  to  a  unit  of 
mass  a  unit  velocity  in  a  unit  of  time.  Therefore  through  the 
use  of  the  air  density  we  can  compare  the  results  of  experiments 
conducted  in  different  units,  or  can  readily  adapt  to  one  system 
a  series  of  measurements  made  in  another  system  of  units. 
We  need  merely  write  the  constant  k  (Ky  or  K*),  used  before, 
in  the  form, 

k=Cp, 

where  p  is  the  density  of  the  air  in  the  units  employed. 
3.  We  shall  take  some  examples. 

(1)  Take  the  meter  as  unit  of  length,  the  second  as  unit 
of  time,  and  the  weight  of  a  kilogram  as  the  unit  of  force.     In 
order  to  have  consistent  units,  the  unit  of  mass  is  then  9.8 
kilograms.     The  density  in  C.G.S  units   of  dry  air  at  15.6°  C. 
(60°  F.),  and  pressure  of  76  cm.  of  mercury  is  .001225.     Hence 
the    mass  of  a  cubic  meter  is  .001225  Xio6/io3  =  1.255   kilo- 
grams.    In  the  system  we  have  adopted  we  have  therefore 
for  the  density  p  =  1.225/9.8  =  .125.     Therefore, 

or    C  =  Sk. 

In  §  i  we  had  k  =  .075  for  the  units  employed  here.  Hence 
C  =  .6oo. 

(2)  Take  the  foot  as  a  unit  of  length,  the  second  as  unit 
of  time,  and  the  pound  as  unit  of  force.     The  unit  of  mass  is 
32.2  Ibs.  (a  slug).     The  density  of  the  air  is  now  .00238.    Hence 

&  =  .002386*,    or    C  =  42o.2&. 

In  §  i  we  had  £  =  .00143  for  the  units  used  here.  Hence 
C  =  .6oi,  which  agrees  with  the  results  given  above  for  metric 
units. 

(3)  Take  the  same  units  for  force  and  length  as  in   (i), 
but  take  the  kilometer  per  hour  as  the  unit  of  velocity.     (This 
is  not  a  consistent  system  of  units.) 

We  write 


APPENDIX  133 

But 

F  =  .i25C4(Fm./sec.)2 

=  .  1 2$CA  (1000/3600  X  V  km./hr.)2 

=  . 009604  (Fkm./hr.)2. 
Hence 

£'  =  .00960    or    0=104.2*'. 

If  we  take  C  =  .6oo  we  have  £'=.0057,  which  is  the  coeffi- 
cient given  in  §  i  for  the  units  in  question. 

(4)  Take  the  same  units  as  in  (2),  except  that  we  express 
velocities  in  miles  per  hour. 
We  write 

F  =  £'.4  (F  mi./hr.)2. 
But 

F  =  . 0023804  (Fft./sec.)2 

=  . 0023804  (5280/3600 X  F  mi./hr.)2 

=  .oo5i204(F  mi./hr.)2. 
Hence 

*'  =  . 005120,    or    0=195.3*'. 

Taking  0  =  .6oo  we  find  £'  =  .0030,  which  agrees  with  the 
value  given  in  §  i. 

As  an  illustration  of  the  use  of  the  results  obtained  we  take 
the  following  problem.  The  lift  coefficient  on  a  certain  wing 
at  an  angle  of  attack  of  4°  is  .001455,  the  units  being  the  pound, 
the  square  foot,  and  mi./hr.  What  is  the  value  of  the  coefficient 
if  the  units  are  the  kilogram,  the  square  meter,  and  the  meter 
per  second? 

We  have  for  the  value  of  the  absolute  constant  by  (4), 
(for  this  wing  at  this  angle  of  attack), 

0=  195.3  X  .001455=  .2841. 

By  (i)  the  value  of  the  lift  coefficient  in  the  new  units  would 
be 

.2841/8  =  .0355. 

(To  make  a  change  of  the  sort  considered  here  we  evidently 
multiply  by  195.3/8  =  24.4.  All  types  of  changes  of  units 


134 


APPENDIX 


likely  to  occur  can  be  similarly  easily  obtained  from  the  results 
above.)* 


II 


4.  In  considering  flight  at  different  altitudes  it  has  been 
necessary  to  consider  the  varying  density  of  the  air.  For  known 
conditions  at  the  surface,  that  is,  known  surface  pressure  and 
temperature,  the  pressure,  density,  and  temperature  at  any 
altitude  can  be  calculated.  The  formulae  for  this  purpose  can 
be  found  in  texts  on  physics,  or  more  extended  aeronautic 
treatises.  We  shall  merely  give  a  table  showing  the  variation 
for  certain  assumed  surface  conditions.  The  ratio  n(z)  is  that 
of  the  pressure  at  altitude  z  to  the  pressure  at  the  surface  of 
the  earth,  and  the  quantity  p(z)  is  the  ratio  of  the  air  density 
at  altitude  z  to  the  surface  density.  It  will  be  seen  that  for 
moderate  altitudes  these  two  quantities  are  approximately 
equal. 


A  l4-lf  11/^A 

TVvrviT-v 

Pressure. 

Density. 

Altitude, 
Feet. 

icinp., 
Fahr. 

Inches, 
Mercury. 

Ratio, 
?(*). 

Slugs. 

Ratio, 
p(«). 

0 

48° 

30.0 

i  .000 

.00246 

i  .000 

2,000 

44 

28.1 

0-937 

.00232 

•940 

4,000 

4i 

26.2 

•873 

.00215 

-875 

6,000 

39 

24.4 

•813 

.00201 

.815 

8,000 

36 

22.6 

•753 

.00187 

.760 

10,000 

30 

2O.9 

.696 

.00175 

.710 

12,000 

25 

JQ-3 

•645 

.00163 

.665 

14,000 

20 

17.9 

.596 

.00151 

•635 

16,000 

12 

16.5 

•550 

.00144 

.600 

18,000 

3 

iS-2 

.508 

.00133 

•  540 

20,000 

-5 

14.0 

.467 

.00125 

•510 

22,000 

—  14 

13.0 

*     -432 

.00118 

.480 

24,000 

-23 

12.  0 

.400 

.00108 

.440 

*  For  a  discussion  of  the  question  of  units  see  Everett's  "Illustrations  of  the 
C.  G.  S.  System  of  Units  with  Tables  of  Physical  Constants,"  Macmillan  &  Co., 
London  and  New  York,  4th  edition,  1891. 


APPENDIX 


135 


III 

5.  The  following  table  is  given  for  the  purpose  cf  changing 
velocities  from  miles  per  hour  to  feet  per  second,  and  con- 
versely. 


mi./hr. 

ft./sec. 

ft./sec. 

mi./hr. 

30 

44.00 

So 

34-09 

40 

58.66 

60 

40.90 

5° 

73-32 

70 

47-72 

60 

88.00 

80 

54-54 

70 

102.66 

90 

61.37 

80 

117.32 

IOO 

68.18 

90 

132.00 

no 

74-99 

IOO 

146.66 

1  20 

81.81 

no 

161.36 

130 

88.62 

140 

95-44 

!     I5° 

102.  26 

IV 

6.  References. — The  literature  on  the  subject  of  Aviation 
has  grown  with  great  rapidity  in  the  past  few  years.  Although 
no  attempt  at  completeness  is  made  in  the  following  list  of 
references  it  is  hoped  that  the  different  aspects  of  the  subject 
are  adequately  covered.  The  title  of  a  work  indicates  whether 
it  is  of  general  or  special  character. 

1.  Cowley  and  Evans,  Aeronautics  in  Theory  and  Experiment, 

Longmans,  Green  &  Co.,  New  York,  1918. 

2.  H.  Shaw,  A  Textbook  on  Aeronautics.  J.  B.  Lippincott  Co., 

Philadelphia,  1919. 

3.  F.  W.  Lancaster,  Aerodynamics. 

4.  G.  Eiffel,  La  Resistance  de  1'Air,  H.  Dunod  et  E.  Pinat, 

Editeurs,  Paris,  1911. 

5.  M.   L.   Legrand,   La  Resistance  de  1'Air,  Librairie  Aero- 

nautique,  40  rue  de  Seine,  Paris. 

6.  A.   See,   Les   Lois   experimentales   de  1'Aviation,  Librairie 

Aeronautique,  Paris. 


136  APPENDIX 

7.  G.  Greenhill,  The  Dynamics  of  Mechanical  Flight,  D.  Van 

Nostrand  Company,  New  York,  1912. 

8.  A.  W.  Judge,  The  Properties  of  Aerofoils  and  Aerodynamic 

Bodies,  James  Selwyn  &  Co.,  London,  1917. 

9.  E.  B.  Wilson,  Aeronautics,  John  Wiley  &  Sons,  Inc.,  1920. 

10.  A  Klemin  and  T.  H.  Huff,  Course  in  Aerodynamics  and 

Aeroplane  Design,  in  Aviation  and  Aeronautical  Engineer- 
ing, Gardner  Moffat  Co.,  New  York,  commencing  in 
Vol.  i,  No.  2,  Aug.  i,  1916;  also  published  separately. 

11.  G.  H.  Bryan,  Stability  in  Aviation,   The  Macmillan  Co., 

London,  1911. 

12.  R.  Devillers,  La  Dynamique  de  P  Avion,  Librairie  Aero- 

nautique,  Paris,  1918. 

13.  G.   De  Bothezat,   Etude   de   la   Stabilite  de   1'Aeroplane, 

H.  Dunod  et  E.  Pinat,  Paris,  1911.  • 

14.  R.    Soreau,   L'Helice   Propulsive,   Librairie   Aeronautique, 

Paris,  1911. 

15.  Duchene,  The  Mechanics  of  the  Aeroplane,  translated  by 

J.  H.  Ledeboer  and  T.  Hubbard,  Longmans,  Green  & 
Co.,  1916. 

1 6.  Smithsonian  Miscellaneous  Collection,  Vol.  62,  1916,  Wash- 

ington, D.  C. 

17.  Reports  of  American  Advisory  Committee  on  Aeronautics. 

1 8.  Reports  of  British  Advisory  Committee  on  Aeronautics. 


INDEX. 


(The  numbers  refer  to  pages) 


Aerofoil,  7. 
Ailerons,  56,  104. 
Altitude,  effect  of,  21,  77. 
Angle,  of  attack,  5. 

economical,  26. 

optimum,  22. 

for  maximum  power  in  ascent, 

45- 
for  maximum  traction  in  ascent, 

45- 
for   minimum    vertical   velocity 

in  ascent,  47. 
for   minimum    vertical   velocity 

in  descent,  40. 
Aspect  ratio,  6. 
Asymmetric  oscillations,  117. 

Bairstow,  121,  122,  123. 
Banking,  angle  of,  53. 
Body  resistance,  15. 
Bryan,  108. 

Camber,  7. 
Cambered  wing,  7. 
Ceiling,  49,  86. 

determination  of,  50,  88. 
Center  of  pressure,  6,  12. 
behavior  of,  6,  13. 
Characteristic  curves  for  wing,  8. 
Chord,  7. 
Circular  descent,  59. 

Descent,  36. 

rectilinear,  33. 
Detrimental  surface,  15. 
Drag,  6. 
Duchemin,  formula  of,  5. 


Economical  angle,  26. 
Efficiency  of  propeller,  77. 
Elevator,  100. 
Equations  of 

ascent,  43. 

circular  descent,  59. 

horizontal  flight,  20, 

stability,  114. 

turning,  53. 
Euler,  108. 

Fineness,  23. 
Fin,  104. 

Gap,  14. 

Gusts,  effect  of,  125. 

Helical  descent,  59. 
Hunsaker,  123,  124. 

Inclination,  in  turning,  54. 
Inherent  stability,  95. 

Landing  speed,  20. 
Leading  edge,  8. 
Lift,  6. 
Loading,  20. 

Metacenter,  98. 
Metacentric  curve,  98. 
Model,  of  wing,  3. 

of  complete  machine,  15. 
Motor  diagram,  81. 
Moving  axes,  104. 


Newton,  formula  of,  5. 
Nose,  8. 


137 


138 


INDEX 


Optimum  angle,  22. 
Oscillation,  asymmetric,  117. 
symmetric,  117. 

Pitch  of  propeller,  71. 

Pitching,  97. 

Polar  diagram,  10. 

Power,  absorbed  by  propeller,  82. 

total,  29. 

used  in  ascent,  46. 

useful,  25. 

Pressure,  on  cambered  wing,  7,  n. 
on  plane,  4,  5. 

Radius  of  action,  90. 
Resistance  derivatives,  116. 
Resistance  of  body,  15. 
Rolling,  97. 

Rotary  derivatives,  116. 
Routh,  108,  109,  106. 
Rudder,  56,  104. 

Stability,  dynamic,  97. 

lateral,  103. 

longitudinal,  97. 

in  rolling,  103. 

static,  96. 
Static  stability,  96. 
Stagger,  14, 


Steady  motion,  115. 
Supercharge,  89. 
Symmetric  oscillations,  117. 
Spiral  instability,  106. 

Tail  plane,  99. 
Thrust  of  propeller,  75. 
Time  of  ascent,  50. 
Total  power,  29. 
Traction,  22,  33,  44. 
Turning,  52. 

Useful  power,  25. 

Velocity  in 

ascent,  44,  47. 
circular  descent,  44,  47, 
descent,  39. 
horizontal  flight,  20, 
turning,  54. 
Vertical  velocity  in 

ascent,  44,  49. 
descent,  39. 
•} 

Wilson,  125. 
Wind  tunnel,  3. 

Yawing,  97. 


^    ^ 

Wiley  Special  Subject  Catalogues 

For  convenience  a  list  of  the  Wiley  Special  Subject 
Catalogues,  envelope  size,  has  been  printed.  These 
are  arranged  in  groups — each  catalogue  having  a  key 
symbol.  (See  special  Subject  List  Below).  To 
obtain  any  of  these  catalogues,  send  a  postal  using 
the  key  symbols  of  the  Catalogues  desired. 

. 

I — Agriculture.     Animal  Husbandry.     Dairying.     Industrial 
Canning  and  Preserving. 

2 — Architecture.  •   Building.      Concrete  and  Masonry. 

3 — Business  Administration  and  Management.     Law. 

Industrial  Processes:   Canning  and  Preserving;    Oil  and  Gas 
Production;  Paint;  Printing;  Sugar  Manufacture;  Textile. 

CHEMISTRY 
'  4a  General;  Analytical,  Qualitative  and  Quantitative;  Inorganic; 

Organic. 
4b  Electro-  and  Physical;  Food  and  Water;  Industrial;  Medical 

and  Pharmaceutical;  Sugar. 

aOI 
CIVIL  ENGINEERING 

5a  Unclassified  and  Structural  Engineering. 

5b  Materials  and  Mechanics  of  Construction,  including;  Cement 
and  Concrete;  Excavation  and  Earthwork;  Foundations; 
Masonry. 

5c  Railroads;  Surveying. 

5d  Dams;  Hydraulic  Engineering;  Pumping  and  Hydraulics;  Irri- 
gation Engineering;  River  and  Harbor  Engineering;  Water 
Supply. 


CIVIL  ENGINEERING—  Continued 

5e  Highways;  Municipal  Engineering;  Sanitary  Engineering; 
Water  Supply.  Forestry.  Horticulture,  Botany  and 
Landscape  Gardening. 


6 — Design.  Decoration.  Drawing:  General;  Descriptive 
Geometry;  Kinematics;  Mechanical. 

ELECTRICAL  ENGINEERING— PHYSICS 

7 — General  and  Unclassified;  Batteries;  Central  Station  Practice; 
Distribution  and  Transmission;  Dynamo-Electro  Machinery; 
Electro-Chemistry  and  Metallurgy;  Measuring  Instruments 
and  Miscellaneous  Apparatus. 

'emi 

£niv'X9£3'x<I  i  >iO 

8 — Astronomy.      Meteorology.      Explosives.      Marine    and 

Naval  Engineering.     Military.     Miscellaneous  Books. 

MATHEMATICS 

9 — General;  Algebra;  Analytic  and  Plane  Geometry;  Calculus; 
Trigonometry;  Vector  Analysis. 

MECHANICAL  ENGINEERING 

lOa  General  and  Unclassified;  Foundry  Practice;  Shop  Practice. 
lOb  Gas  Power  and    Internal   Combustion  Engines;  Heating  and 

Ventilation;  Refrigeration. 
lOc   Machine  Design  and  Mechanism;  Power  Transmission;  Steam 

Power  and  Power  Plants;  Thermodynamics  and  Heat  Power. 

11— Mechanics.  

12 — Medicine.  Pharmacy.  Medical  and  Pharmaceutical  Chem- 
istry. Sanitary  Science  and  Engineering.  Bacteriology  and 

Biology. 

MINING  ENGINEERING 

13 — General;  Assaying;  Excavation,  Earthwork,  Tunneling,  Etc.; 
Explosives;  Geology;  Metallurgy;  Mineralogy;  Prospecting^: 
Ventilation. 

14 — Food  and  Water.  Sanitation.  Landscape  Gardening. 
Design  and  Decoration.  Housing,  House  Painting. 


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